Sr Examen
Integral de √(r^2-y^2) dy
Solución
Ha introducido
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1 / | | _________ | / 2 2 | \/ r - y dy | / 0
$$\int\limits_{0}^{1} \sqrt{r^{2} - y^{2}}\, dy$$
Integral(sqrt(r^2 - y^2), (y, 0, 1))
Respuesta (Indefinida)
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// 2 /y\ \
|| I*r *acosh|-| 3 | 2| |
|| \r/ I*y I*r*y |y | |
||- ------------- + ------------------- - ----------------- for |--| > 1|
|| 2 _________ _________ | 2| |
/ || / 2 / 2 |r | |
| || / y / y |
| _________ || 2*r* / -1 + -- 2* / -1 + -- |
| / 2 2 || / 2 / 2 |
| \/ r - y dy = C + |< \/ r \/ r |
| || |
/ || ________ |
|| / 2 |
|| / y |
|| 2 /y\ r*y* / 1 - -- |
|| r *asin|-| / 2 |
|| \r/ \/ r |
|| ---------- + ------------------ otherwise |
\\ 2 2 /
$$\int \sqrt{r^{2} - y^{2}}\, dy = C + \begin{cases} - \frac{i r^{2} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{2} - \frac{i r y}{2 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{3}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{2} \operatorname{asin}{\left(\frac{y}{r} \right)}}{2} + \frac{r y \sqrt{1 - \frac{y^{2}}{r^{2}}}}{2} & \text{otherwise} \end{cases}$$
Respuesta
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1 / | | / 2 4 2 2 | | I*r I*y I*y 3*I*y y | |- --------------- + ---------------- - ----------------- + ------------------- for ---- > 1 | | _________ 3/2 3/2 _________ | 2| | | / 2 / 2\ / 2\ / 2 |r | | | / y | y | 3 | y | / y | | / -1 + -- 2*r*|-1 + --| 2*r *|-1 + --| 2*r* / -1 + -- | | / 2 | 2| | 2| / 2 | | \/ r \ r / \ r / \/ r | | | | ________ | < / 2 dy | | / y | | r* / 1 - -- | | / 2 2 | | \/ r r y | | ---------------- + ---------------- - ------------------ otherwise | | 2 ________ ________ | | / 2 / 2 | | / y / y | | 2* / 1 - -- 2*r* / 1 - -- | | / 2 / 2 | \ \/ r \/ r | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i r}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{3 i y^{2}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{2}}{2 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{i y^{4}}{2 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{y^{2}}{\left|{r^{2}}\right|} > 1 \\\frac{r \sqrt{1 - \frac{y^{2}}{r^{2}}}}{2} + \frac{r}{2 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{y^{2}}{2 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy$$
=
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1 / | | / 2 4 2 2 | | I*r I*y I*y 3*I*y y | |- --------------- + ---------------- - ----------------- + ------------------- for ---- > 1 | | _________ 3/2 3/2 _________ | 2| | | / 2 / 2\ / 2\ / 2 |r | | | / y | y | 3 | y | / y | | / -1 + -- 2*r*|-1 + --| 2*r *|-1 + --| 2*r* / -1 + -- | | / 2 | 2| | 2| / 2 | | \/ r \ r / \ r / \/ r | | | | ________ | < / 2 dy | | / y | | r* / 1 - -- | | / 2 2 | | \/ r r y | | ---------------- + ---------------- - ------------------ otherwise | | 2 ________ ________ | | / 2 / 2 | | / y / y | | 2* / 1 - -- 2*r* / 1 - -- | | / 2 / 2 | \ \/ r \/ r | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i r}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{3 i y^{2}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{2}}{2 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{i y^{4}}{2 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{y^{2}}{\left|{r^{2}}\right|} > 1 \\\frac{r \sqrt{1 - \frac{y^{2}}{r^{2}}}}{2} + \frac{r}{2 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{y^{2}}{2 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy$$
Integral(Piecewise((-i*r/sqrt(-1 + y^2/r^2) + i*y^2/(2*r*(-1 + y^2/r^2)^(3/2)) - i*y^4/(2*r^3*(-1 + y^2/r^2)^(3/2)) + 3*i*y^2/(2*r*sqrt(-1 + y^2/r^2)), y^2/|r^2| > 1), (r*sqrt(1 - y^2/r^2)/2 + r/(2*sqrt(1 - y^2/r^2)) - y^2/(2*r*sqrt(1 - y^2/r^2)), True)), (y, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.