1 / | | _____________ | / 2 2 | \/ 1 - x - y dy | / 0
Integral(sqrt(1 - x^2 - y^2), (y, 0, 1))
// _________________________ _________________________ \ || _________________________ _________________________ / 2 / 2 | || / 2 / 2 / y / 2\ / y \ 2 / y / 2\ / y \ | || / y / 2\ 2 / y / 2\ 2*I* / -1 + ------------------ *\1 - x /*acosh|-----------------------| 2*I*x * / -1 + ------------------ *\1 - x /*acosh|-----------------------| | || pi* / -1 + ------------------ *\1 - x / ____________________ ____________________ pi*x * / -1 + ------------------ *\1 - x / ____________________ / / 2\ | ____________________| / / 2\ | ____________________| | || / / 2\ 3 / / 2\ / / 2\ / / 2\ 2 / / 2\ \/ polar_lift\1 - x / | / / 2\ | \/ polar_lift\1 - x / | / / 2\ | | 2 | | || \/ polar_lift\1 - x / 2*I*y *\/ polar_lift\1 - x / 2*I*y*\/ polar_lift\1 - x / \/ polar_lift\1 - x / 2*I*y*x *\/ polar_lift\1 - x / \\/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / | y | | ||- -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + -------------------------------------------------------------------------- + -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + --------------------------------------------------------------------------- - ------------------------------------------------------------------------------ for |-------| > 1| || _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ | 2| | || / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 |-1 + x | | / || / y 2 / y / y 2 / y / y 2 / y / y 2 / y / y 2 / y / y 2 / y / y 2 / y | | || - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | | _____________ || / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ | | / 2 2 || \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / | | \/ 1 - x - y dy = C + |< | | || ________________________ ________________________ | / || / 2 / 2 | || / y / 2\ / y \ 2 / y / 2\ / y \ | || / 1 - ------------------ *\1 - x /*asin|-----------------------| x * / 1 - ------------------ *\1 - x /*asin|-----------------------| | || ____________________ ____________________ ____________________ / / 2\ | ____________________| / / 2\ | ____________________| | || 3 / / 2\ / / 2\ 2 / / 2\ \/ polar_lift\1 - x / | / / 2\ | \/ polar_lift\1 - x / | / / 2\ | | || y *\/ polar_lift\1 - x / y*\/ polar_lift\1 - x / y*x *\/ polar_lift\1 - x / \\/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / | || ------------------------------------------------------------------------ - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ otherwise | || ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ | || / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 | || / y 2 / y / y 2 / y / y 2 / y / y 2 / y / y 2 / y | || - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | || / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ | \\ \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / /
1 / | | / _________________________ _________________________ _________________________ _________________________ | | / 2 / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | / y / 2\ | 4*y 4*y*x | 3 / / 2\ | 4*y 4*y*x | / / 2\ | 4*y 4*y*x | 2 / y / 2\ | 4*y 4*y*x | 2 / / 2\ | 4*y 4*y*x | / y / 2\ | 4*y 4*y*x | / y \ 2 / y / 2\ | 4*y 4*y*x | / y \ | | pi* / -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------| 2*I*y *\/ polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------| 2*I*y*\/ polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------| pi*x * / -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------| 2*I*y*x *\/ polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------| 2*I* / -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------| 2*I*x * / -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------| | | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | ____________________| / / 2\ | _________________________ _________________________ | | ____________________| | | \/ polar_lift\1 - x / | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - x / | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | / 2\ / y \ 2 / 2\ / y \ | | | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| \\/ polar_lift\1 - x / / | / y / 2\ / y / 2\| \\/ polar_lift\1 - x / / 2*I*y*\1 - x /*acosh|-----------------------| 2*I*y*x *\1 - x /*acosh|-----------------------| | | ____________________ ____________________ ____________________ | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | ____________________| | ____________________| | | / / 2\ 2 / / 2\ 2 / / 2\ / 2\ | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ | | / / 2\ | 2 | | 2*I*\/ polar_lift\1 - x / 6*I*y *\/ polar_lift\1 - x / 2*I*x *\/ polar_lift\1 - x / 2*I*\1 - x / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / 2*I*x *\1 - x / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / pi*y*\1 - x / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / pi*y*x *\1 - x / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / y | |-------------------------------------------------------------------------- - -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- for --------- > 1 | | _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ / _________________________ _________________________\ 2 / _________________________ _________________________\ 2 2 2 _________________________ / _________________________ _________________________\ 2 2 _________________________ / _________________________ _________________________\ 2 _________________________ / _________________________ _________________________\ _________________________ / _________________________ _________________________\ | 2| | | / 2 / 2 / 2 / 2 / 2 / 2 | / 2 / 2 | ____________________ / _________________________ _________________________\ | / 2 / 2 | ____________________ / _________________________ _________________________\ / _________________________ _________________________\ / _________________________ _________________________\ / 2 | / 2 / 2 | / _________________________ _________________________\ / _________________________ _________________________\ / 2 | / 2 / 2 | / _________________________ _________________________\ / 2 | / 2 / 2 | / 2 | / 2 / 2 | |-1 + x | | | / y 2 / y / y 2 / y / y 2 / y | / y 2 / y | / / 2\ | / 2 / 2 | | / y 2 / y | / / 2\ | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / y | / y 2 / y | / 2\ | / 2 / 2 | | / 2 / 2 | / y | / y 2 / y | / 2\ | / 2 / 2 | / y | / y 2 / y | / 2\ / y | / y 2 / y | / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*\/ polar_lift\1 - x / | / y 2 / y | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*\/ polar_lift\1 - x / | / y 2 / y | | / y 2 / y | | / y 2 / y | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*polar_lift\1 - x / | / y 2 / y | | / y 2 / y | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*polar_lift\1 - x / | / y 2 / y | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*polar_lift\1 - x / / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*polar_lift\1 - x / | | / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ | | | \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | | \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | < dy | | ________________________ ________________________ | | ____________________ / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | 3 / / 2\ | 2*y 2*y*x | / / 2\ | 2*y 2*y*x | 2 / / 2\ | 2*y 2*y*x | / y / 2\ | 2*y 2*y*x | / y \ 2 / y / 2\ | 2*y 2*y*x | / y \ | | y *\/ polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------| y*\/ polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------| y*x *\/ polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------| / 1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------| x * / 1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------| | | | ________________________ ________________________ | | ________________________ ________________________ | | ________________________ ________________________ | / / 2\ | ________________________ ________________________ | | ____________________| / / 2\ | ________________________ ________________________ | | ____________________| | | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | / 2\ / y \ \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | 2 / 2\ / y \ | | | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| \\/ polar_lift\1 - x / / y*\1 - x /*asin|-----------------------| | / y / 2\ / y / 2\| \\/ polar_lift\1 - x / / y*x *\1 - x /*asin|-----------------------| | | ____________________ ____________________ ____________________ | / 1 - ------------------ *polar_lift\1 - x / / 1 - ------------------ *polar_lift\1 - x /| | / 1 - ------------------ *polar_lift\1 - x / / 1 - ------------------ *polar_lift\1 - x /| | / 1 - ------------------ *polar_lift\1 - x / / 1 - ------------------ *polar_lift\1 - x /| | / 1 - ------------------ *polar_lift\1 - x / / 1 - ------------------ *polar_lift\1 - x /| | ____________________| | / 1 - ------------------ *polar_lift\1 - x / / 1 - ------------------ *polar_lift\1 - x /| | ____________________| | | / / 2\ 2 / / 2\ 2 2 / / 2\ 2 / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ | | / / 2\ / / 2\ | | / / 2\ | | | \/ polar_lift\1 - x / x *\/ polar_lift\1 - x / 1 - x 3*y *\/ polar_lift\1 - x / x *\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / | | - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------- otherwise | | ________________________ ________________________ ________________________ ________________________ / ________________________ ________________________\ ________________________ ________________________ / ________________________ ________________________\ 2 2 2 2 ________________________ / ________________________ ________________________\ 2 ________________________ / ________________________ ________________________\ | | / 2 / 2 / 2 / 2 | / 2 / 2 | ____________________ / 2 / 2 | / 2 / 2 | ____________________ / ________________________ ________________________\ / ________________________ ________________________\ / ________________________ ________________________\ / ________________________ ________________________\ / 2 | / 2 / 2 | / ________________________ ________________________\ / 2 | / 2 / 2 | | | / y 2 / y / y 2 / y | / y 2 / y | / / 2\ / y 2 / y | / y 2 / y | / / 2\ | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / y | / y 2 / y | / 2\ | / 2 / 2 | / y | / y 2 / y | / 2\ | | - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |*\/ polar_lift\1 - x / - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |*\/ polar_lift\1 - x / | / y 2 / y | | / y 2 / y | | / y 2 / y | | / y 2 / y | / 1 - ------------------ *|- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |*polar_lift\1 - x / | / y 2 / y | / 1 - ------------------ *|- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |*polar_lift\1 - x / | | / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | / / 2\ / / 2\ | / / 2\ / / 2\ | |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | | | \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | \ \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / 0
=
1 / | | / _________________________ _________________________ _________________________ _________________________ | | / 2 / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | / y / 2\ | 4*y 4*y*x | 3 / / 2\ | 4*y 4*y*x | / / 2\ | 4*y 4*y*x | 2 / y / 2\ | 4*y 4*y*x | 2 / / 2\ | 4*y 4*y*x | / y / 2\ | 4*y 4*y*x | / y \ 2 / y / 2\ | 4*y 4*y*x | / y \ | | pi* / -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------| 2*I*y *\/ polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------| 2*I*y*\/ polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------| pi*x * / -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------| 2*I*y*x *\/ polar_lift\1 - x / *|-------------------------------------------------- - --------------------------------------------------| 2*I* / -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------| 2*I*x * / -1 + ------------------ *\1 - x /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------| | | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | _________________________ _________________________ | / / 2\ | _________________________ _________________________ | | ____________________| / / 2\ | _________________________ _________________________ | | ____________________| | | \/ polar_lift\1 - x / | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - x / | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | / 2\ / y \ 2 / 2\ / y \ | | | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| \\/ polar_lift\1 - x / / | / y / 2\ / y / 2\| \\/ polar_lift\1 - x / / 2*I*y*\1 - x /*acosh|-----------------------| 2*I*y*x *\1 - x /*acosh|-----------------------| | | ____________________ ____________________ ____________________ | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | / -1 + ------------------ *polar_lift\1 - x / / -1 + ------------------ *polar_lift\1 - x /| | ____________________| | ____________________| | | / / 2\ 2 / / 2\ 2 / / 2\ / 2\ | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | 2 / 2\ | / / 2\ / / 2\ | | / / 2\ | | / / 2\ | 2 | | 2*I*\/ polar_lift\1 - x / 6*I*y *\/ polar_lift\1 - x / 2*I*x *\/ polar_lift\1 - x / 2*I*\1 - x / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / 2*I*x *\1 - x / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / pi*y*\1 - x / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / pi*y*x *\1 - x / \\/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / y | |-------------------------------------------------------------------------- - -------------------------------------------------------------------------- - -------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------- for --------- > 1 | | _________________________ _________________________ _________________________ _________________________ _________________________ _________________________ / _________________________ _________________________\ 2 / _________________________ _________________________\ 2 2 2 _________________________ / _________________________ _________________________\ 2 2 _________________________ / _________________________ _________________________\ 2 _________________________ / _________________________ _________________________\ _________________________ / _________________________ _________________________\ | 2| | | / 2 / 2 / 2 / 2 / 2 / 2 | / 2 / 2 | ____________________ / _________________________ _________________________\ | / 2 / 2 | ____________________ / _________________________ _________________________\ / _________________________ _________________________\ / _________________________ _________________________\ / 2 | / 2 / 2 | / _________________________ _________________________\ / _________________________ _________________________\ / 2 | / 2 / 2 | / _________________________ _________________________\ / 2 | / 2 / 2 | / 2 | / 2 / 2 | |-1 + x | | | / y 2 / y / y 2 / y / y 2 / y | / y 2 / y | / / 2\ | / 2 / 2 | | / y 2 / y | / / 2\ | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / y | / y 2 / y | / 2\ | / 2 / 2 | | / 2 / 2 | / y | / y 2 / y | / 2\ | / 2 / 2 | / y | / y 2 / y | / 2\ / y | / y 2 / y | / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ - 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*\/ polar_lift\1 - x / | / y 2 / y | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*\/ polar_lift\1 - x / | / y 2 / y | | / y 2 / y | | / y 2 / y | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*polar_lift\1 - x / | / y 2 / y | | / y 2 / y | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*polar_lift\1 - x / | / y 2 / y | / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*polar_lift\1 - x / / -1 + ------------------ *|- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ |*polar_lift\1 - x / | | / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | |- 4* / -1 + ------------------ + 4*x * / -1 + ------------------ | / / 2\ | / / 2\ / / 2\ | / / 2\ | / / 2\ / / 2\ | | | \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | | \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | < dy | | ________________________ ________________________ | | ____________________ / 2 \ ____________________ / 2 \ ____________________ / 2 \ / 2 / 2 \ / 2 / 2 \ | | 3 / / 2\ | 2*y 2*y*x | / / 2\ | 2*y 2*y*x | 2 / / 2\ | 2*y 2*y*x | / y / 2\ | 2*y 2*y*x | / y \ 2 / y / 2\ | 2*y 2*y*x | / y \ | | y *\/ polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------| y*\/ polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------| y*x *\/ polar_lift\1 - x / *|- ------------------------------------------------- + -------------------------------------------------| / 1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------| x * / 1 - ------------------ *\1 - x /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------| | | | ________________________ ________________________ | | ________________________ ________________________ | | ________________________ ________________________ | / / 2\ | ________________________ ________________________ | | ____________________| / / 2\ | ________________________ ________________________ | | ____________________| | | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | / 2\ / y \ \/ polar_lift\1 - x / | / 2 / 2 | | / / 2\ | 2 / 2\ / y \ | | | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| | / y / 2\ / y / 2\| \\/ polar_lift\1 - x / / y*\1 - x /*asin|-----------------------| | / y / 2\ / y / 2\| \\/ polar_lift\1 - x / / y*x *\1 - x /*asin|-----------------------| | | ____________________ ____________________ ____________________ | / 1 - ------------------ *polar_lift\1 - x / / 1 - ------------------ *polar_lift\1 - x /| | / 1 - ------------------ *polar_lift\1 - x / / 1 - ------------------ *polar_lift\1 - x /| | / 1 - ------------------ *polar_lift\1 - x / / 1 - ------------------ *polar_lift\1 - x /| | / 1 - ------------------ *polar_lift\1 - x / / 1 - ------------------ *polar_lift\1 - x /| | ____________________| | / 1 - ------------------ *polar_lift\1 - x / / 1 - ------------------ *polar_lift\1 - x /| | ____________________| | | / / 2\ 2 / / 2\ 2 2 / / 2\ 2 / 2\ | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ | | / / 2\ / / 2\ | | / / 2\ | | | \/ polar_lift\1 - x / x *\/ polar_lift\1 - x / 1 - x 3*y *\/ polar_lift\1 - x / x *\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \\/ polar_lift\1 - x / / | | - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------- otherwise | | ________________________ ________________________ ________________________ ________________________ / ________________________ ________________________\ ________________________ ________________________ / ________________________ ________________________\ 2 2 2 2 ________________________ / ________________________ ________________________\ 2 ________________________ / ________________________ ________________________\ | | / 2 / 2 / 2 / 2 | / 2 / 2 | ____________________ / 2 / 2 | / 2 / 2 | ____________________ / ________________________ ________________________\ / ________________________ ________________________\ / ________________________ ________________________\ / ________________________ ________________________\ / 2 | / 2 / 2 | / ________________________ ________________________\ / 2 | / 2 / 2 | | | / y 2 / y / y 2 / y | / y 2 / y | / / 2\ / y 2 / y | / y 2 / y | / / 2\ | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | | / 2 / 2 | / y | / y 2 / y | / 2\ | / 2 / 2 | / y | / y 2 / y | / 2\ | | - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |*\/ polar_lift\1 - x / - 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |*\/ polar_lift\1 - x / | / y 2 / y | | / y 2 / y | | / y 2 / y | | / y 2 / y | / 1 - ------------------ *|- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |*polar_lift\1 - x / | / y 2 / y | / 1 - ------------------ *|- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ |*polar_lift\1 - x / | | / / 2\ / / 2\ / / 2\ / / 2\ | / / 2\ / / 2\ | / / 2\ / / 2\ | / / 2\ / / 2\ | |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | |- 2* / 1 - ------------------ + 2*x * / 1 - ------------------ | / / 2\ | / / 2\ / / 2\ | | | \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \/ polar_lift\1 - x / \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | | / / 2\ / / 2\ | \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / / 2\ / / 2\ | \/ polar_lift\1 - x / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | \ \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / \ \/ polar_lift\1 - x / \/ polar_lift\1 - x / / | / 0
Integral(Piecewise((2*i*sqrt(polar_lift(1 - x^2))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))) - 6*i*y^2*sqrt(polar_lift(1 - x^2))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))) - 2*i*x^2*sqrt(polar_lift(1 - x^2))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))) + 2*i*(1 - x^2)/((-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) - pi*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 - 2*i*x^2*(1 - x^2)/((-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) - 2*i*y^3*sqrt(polar_lift(1 - x^2))*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + 2*i*y*sqrt(polar_lift(1 - x^2))*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + pi*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 - pi*y*(1 - x^2)/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) - 2*i*y*x^2*sqrt(polar_lift(1 - x^2))*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + 2*i*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*acosh(y/sqrt(polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + pi*y*x^2*(1 - x^2)/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) - 2*i*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2))*(1 - x^2)*(4*y/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) - 4*y*x^2/(sqrt(-1 + y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*acosh(y/sqrt(polar_lift(1 - x^2)))/(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))^2 + 2*i*y*(1 - x^2)*acosh(y/sqrt(polar_lift(1 - x^2)))/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) - 2*i*y*x^2*(1 - x^2)*acosh(y/sqrt(polar_lift(1 - x^2)))/(sqrt(-1 + y^2/polar_lift(1 - x^2))*(-4*sqrt(-1 + y^2/polar_lift(1 - x^2)) + 4*x^2*sqrt(-1 + y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)), y^2/|-1 + x^2| > 1), (-sqrt(polar_lift(1 - x^2))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))) + x^2*sqrt(polar_lift(1 - x^2))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))) - (1 - x^2)/((-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) + 3*y^2*sqrt(polar_lift(1 - x^2))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2))) + x^2*(1 - x^2)/((-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*sqrt(polar_lift(1 - x^2))) + y^3*sqrt(polar_lift(1 - x^2))*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 - y*sqrt(polar_lift(1 - x^2))*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 + y*x^2*sqrt(polar_lift(1 - x^2))*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 - sqrt(1 - y^2/polar_lift(1 - x^2))*(1 - x^2)*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*asin(y/sqrt(polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 + y*(1 - x^2)*asin(y/sqrt(polar_lift(1 - x^2)))/(sqrt(1 - y^2/polar_lift(1 - x^2))*(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)) + x^2*sqrt(1 - y^2/polar_lift(1 - x^2))*(1 - x^2)*(-2*y/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)) + 2*y*x^2/(sqrt(1 - y^2/polar_lift(1 - x^2))*polar_lift(1 - x^2)))*asin(y/sqrt(polar_lift(1 - x^2)))/(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))^2 - y*x^2*(1 - x^2)*asin(y/sqrt(polar_lift(1 - x^2)))/(sqrt(1 - y^2/polar_lift(1 - x^2))*(-2*sqrt(1 - y^2/polar_lift(1 - x^2)) + 2*x^2*sqrt(1 - y^2/polar_lift(1 - x^2)))*polar_lift(1 - x^2)), True)), (y, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.