6.2 Volumes/15: Difference between revisions
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(Created page with "<math> x=y^2, x=1; \text{about x=1} </math> <math> \begin{align} \pi\int_-1^1\left(1-y^2\right)^2 dy &= \pi\int_-1^1\left(1-2y^2+y^4\right)dy \\[2ex] &= \pi\left[y-\frac{2}{3}y^3+\frac{y^5}{5}\right]\Bigg|_-1^1 \\[2ex] &= \pi\left[1+\frac{2}{3}+\frac{1}{5}\right]- \pileft[-1+\frac{2}{5}-\frac{1}{5}\right] \\[2ex] &= \frac{16\pi}{15} \end{align} </math>") |
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<math> | <math> | ||
\begin{align} | \begin{align} | ||
\pi\int_-1^1\left(1-y^2\right)^2 dy &= \pi\int_-1^1\left(1-2y^2+y^4\right)dy \\[2ex] | \pi\int_{-1}^{1}\left(1-y^2\right)^{2} dy & = \pi\int_{-1}^{1}\left(1-2y^2+y^4\right)dy \\[2ex] | ||
&= \pi\left[y-\frac{2}{3}y^3+\frac{y^5}{5}\right]\Bigg|_-1^1 \\[2ex] | &= \pi\left[y-\frac{2}{3}y^3+\frac{y^5}{5}\right]\Bigg|_{-1}^{1} \\[2ex] | ||
&= \pi\left[1+\frac{2}{3}+\frac{1}{5}\right]- \ | &= \pi\left[1+\frac{2}{3}+\frac{1}{5}\right]- \pi\left[-1+\frac{2}{5}-\frac{1}{5}\right] \\[2ex] | ||
&= \frac{16\pi}{15} | &= \frac{16\pi}{15} | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
Latest revision as of 17:14, 13 December 2022
![{\displaystyle {\begin{aligned}\pi \int _{-1}^{1}\left(1-y^{2}\right)^{2}dy&=\pi \int _{-1}^{1}\left(1-2y^{2}+y^{4}\right)dy\\[2ex]&=\pi \left[y-{\frac {2}{3}}y^{3}+{\frac {y^{5}}{5}}\right]{\Bigg |}_{-1}^{1}\\[2ex]&=\pi \left[1+{\frac {2}{3}}+{\frac {1}{5}}\right]-\pi \left[-1+{\frac {2}{5}}-{\frac {1}{5}}\right]\\[2ex]&={\frac {16\pi }{15}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7974c6d7507cf74484b0e00c3a9e2eed715d633e)