Sr Examen
Integral de 1/(4r^2-x^2-y^2)^(1/2) dy
Solución
Ha introducido
[src]
______________
\/ x*(-x + 2*r)
/
|
| 1
| ------------------- dy
| ________________
| / 2 2 2
| \/ 4*r - x - y
|
/
0
$$\int\limits_{0}^{\sqrt{x \left(2 r - x\right)}} \frac{1}{\sqrt{- y^{2} + \left(4 r^{2} - x^{2}\right)}}\, dy$$
Integral(1/(sqrt(4*r^2 - x^2 - y^2)), (y, 0, sqrt(x*(-x + 2*r))))
Respuesta (Indefinida)
[src]
// | 2 | \
|| / y \ | y | |
||-I*acosh|----------------------------| for |-----------| > 1|
/ || | _________________________| | 2 2| |
| || | / / 2 2\ | |- x + 4*r | |
| 1 || \\/ polar_lift\- x + 4*r / / |
| ------------------- dy = C + |< |
| ________________ || / y \ |
| / 2 2 2 || asin|----------------------------| otherwise |
| \/ 4*r - x - y || | _________________________| |
| || | / / 2 2\ | |
/ || \\/ polar_lift\- x + 4*r / / |
\\ /
$$\int \frac{1}{\sqrt{- y^{2} + \left(4 r^{2} - x^{2}\right)}}\, dy = C + \begin{cases} - i \operatorname{acosh}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}}} \right)} & \text{for}\: \left|{\frac{y^{2}}{4 r^{2} - x^{2}}}\right| > 1 \\\operatorname{asin}{\left(\frac{y}{\sqrt{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}}} \right)} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 0 | / | | | | / | 2 | | | | -I | y | | | |----------------------------------------------------------------- for |-----------| > 1 | | | ______________________________ | 2 2| | | | / 2 _________________________ |- x + 4*r | | | | / y / / 2 2\ | | | / -1 + ----------------------- *\/ polar_lift\- x + 4*r / | | | / / 2 2\ | | |\/ polar_lift\- x + 4*r / ______________ |- | < dy for \/ x*(-x + 2*r) < 0 | | | 1 | | |---------------------------------------------------------------- otherwise | | | _____________________________ | | | / 2 _________________________ | | | / y / / 2 2\ | | | / 1 - ----------------------- *\/ polar_lift\- x + 4*r / | | | / / 2 2\ | | |\/ polar_lift\- x + 4*r / | | \ | | | / | ______________ | \/ x*(-x + 2*r) < | ______________ | \/ x*(-x + 2*r) | / | | | | / | 2 | | | | -I | y | | | |----------------------------------------------------------------- for |-----------| > 1 | | | ______________________________ | 2 2| | | | / 2 _________________________ |- x + 4*r | | | | / y / / 2 2\ | | | / -1 + ----------------------- *\/ polar_lift\- x + 4*r / | | | / / 2 2\ | | |\/ polar_lift\- x + 4*r / | | < dy otherwise | | | 1 | | |---------------------------------------------------------------- otherwise | | | _____________________________ | | | / 2 _________________________ | | | / y / / 2 2\ | | | / 1 - ----------------------- *\/ polar_lift\- x + 4*r / | | | / / 2 2\ | | |\/ polar_lift\- x + 4*r / | | \ | | | / \ 0
$$\begin{cases} - \int\limits_{\sqrt{x \left(2 r - x\right)}}^{0} \begin{cases} - \frac{i}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}}} & \text{for}\: \left|{\frac{y^{2}}{4 r^{2} - x^{2}}}\right| > 1 \\\frac{1}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}}} & \text{otherwise} \end{cases}\, dy & \text{for}\: \sqrt{x \left(2 r - x\right)} < 0 \\\int\limits_{0}^{\sqrt{x \left(2 r - x\right)}} \begin{cases} - \frac{i}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}}} & \text{for}\: \left|{\frac{y^{2}}{4 r^{2} - x^{2}}}\right| > 1 \\\frac{1}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}}} & \text{otherwise} \end{cases}\, dy & \text{otherwise} \end{cases}$$
=
=
/ 0 | / | | | | / | 2 | | | | -I | y | | | |----------------------------------------------------------------- for |-----------| > 1 | | | ______________________________ | 2 2| | | | / 2 _________________________ |- x + 4*r | | | | / y / / 2 2\ | | | / -1 + ----------------------- *\/ polar_lift\- x + 4*r / | | | / / 2 2\ | | |\/ polar_lift\- x + 4*r / ______________ |- | < dy for \/ x*(-x + 2*r) < 0 | | | 1 | | |---------------------------------------------------------------- otherwise | | | _____________________________ | | | / 2 _________________________ | | | / y / / 2 2\ | | | / 1 - ----------------------- *\/ polar_lift\- x + 4*r / | | | / / 2 2\ | | |\/ polar_lift\- x + 4*r / | | \ | | | / | ______________ | \/ x*(-x + 2*r) < | ______________ | \/ x*(-x + 2*r) | / | | | | / | 2 | | | | -I | y | | | |----------------------------------------------------------------- for |-----------| > 1 | | | ______________________________ | 2 2| | | | / 2 _________________________ |- x + 4*r | | | | / y / / 2 2\ | | | / -1 + ----------------------- *\/ polar_lift\- x + 4*r / | | | / / 2 2\ | | |\/ polar_lift\- x + 4*r / | | < dy otherwise | | | 1 | | |---------------------------------------------------------------- otherwise | | | _____________________________ | | | / 2 _________________________ | | | / y / / 2 2\ | | | / 1 - ----------------------- *\/ polar_lift\- x + 4*r / | | | / / 2 2\ | | |\/ polar_lift\- x + 4*r / | | \ | | | / \ 0
$$\begin{cases} - \int\limits_{\sqrt{x \left(2 r - x\right)}}^{0} \begin{cases} - \frac{i}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}}} & \text{for}\: \left|{\frac{y^{2}}{4 r^{2} - x^{2}}}\right| > 1 \\\frac{1}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}}} & \text{otherwise} \end{cases}\, dy & \text{for}\: \sqrt{x \left(2 r - x\right)} < 0 \\\int\limits_{0}^{\sqrt{x \left(2 r - x\right)}} \begin{cases} - \frac{i}{\sqrt{\frac{y^{2}}{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}}} & \text{for}\: \left|{\frac{y^{2}}{4 r^{2} - x^{2}}}\right| > 1 \\\frac{1}{\sqrt{- \frac{y^{2}}{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(4 r^{2} - x^{2} \right)}}} & \text{otherwise} \end{cases}\, dy & \text{otherwise} \end{cases}$$
Piecewise((-Integral(Piecewise((-i/(sqrt(-1 + y^2/polar_lift(-x^2 + 4*r^2))*sqrt(polar_lift(-x^2 + 4*r^2))), Abs(y^2/(-x^2 + 4*r^2)) > 1), (1/(sqrt(1 - y^2/polar_lift(-x^2 + 4*r^2))*sqrt(polar_lift(-x^2 + 4*r^2))), True)), (y, sqrt(x*(-x + 2*r)), 0)), sqrt(x*(-x + 2*r)) < 0), (Integral(Piecewise((-i/(sqrt(-1 + y^2/polar_lift(-x^2 + 4*r^2))*sqrt(polar_lift(-x^2 + 4*r^2))), Abs(y^2/(-x^2 + 4*r^2)) > 1), (1/(sqrt(1 - y^2/polar_lift(-x^2 + 4*r^2))*sqrt(polar_lift(-x^2 + 4*r^2))), True)), (y, 0, sqrt(x*(-x + 2*r)))), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.