【微分積分】8-2-3 円の中心が原点以外の極座標変換｜問題集

\(D\)を極座標変換すると
\(\displaystyle E=\{(r,\theta)|0\leqq r\leqq 2,0\leqq \theta\leqq 2\pi\}\)
よって
\(\displaystyle \iint_D(x^2+y^2)dxdy\)
\(\displaystyle =\iint_E\{(1+r\cos\theta)^2+(1+r\sin\theta)^2\}・rdrd\theta\)
\(\displaystyle =\int_0^2\int_0^{2\pi}\{r^3+2r+2r^2(\cos\theta+\sin\theta)\}d\theta dr\)
\(\displaystyle =2\pi\int_0^2(r^3+2r)dr\)
\(\displaystyle =2\pi\left[\frac{r^4}{4}+r^2\right]_0^2\)
\(=16\pi\)

\(D\)を極座標変換すると
\(\displaystyle E=\{(r,\theta)|0\leqq r\leqq 2\cos\theta,-\frac{\pi}{2}\leqq \theta\leqq \frac{\pi}{2}\}\)
よって
\(\displaystyle \iint_D\sqrt{4-x^2-y^2}dxdy\)
\(\displaystyle =\iint_E\sqrt{4-r^2}・rdrd\theta\)
\(\displaystyle =\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\int_0^{2\cos\theta}r\sqrt{4-r^2}drd\theta\)
\(\displaystyle =\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\frac{8}{3}(1-|\sin\theta|^3)d\theta\)
\(\displaystyle =\frac{8}{3}\pi-\frac{16}{3}\int_0^\frac{\pi}{2}\sin^3\theta d\theta\)
\(\displaystyle =\frac{8}{3}\pi-\frac{32}{9}\)

\(D\)を極座標変換すると
\(\displaystyle E=\{(r,\theta)|2\cos\theta\leqq r\leqq 2,0\leqq \theta\leqq \frac{\pi}{2}\}\)
よって
\(\displaystyle \iint_D\sqrt{x^2+y^2}dxdy\)
\(\displaystyle =\iint_Er・rdrd\theta\)
\(\displaystyle =\int_0^\frac{\pi}{2}\int_{2\cos\theta}^2r^2drd\theta\)
\(\displaystyle =\frac{8}{3}\int_0^\frac{\pi}{2}(1-\cos^3\theta)d\theta\)
\(\displaystyle =\frac{8}{3}\left(\frac{\pi}{2}-\frac{2}{3}\right)\)
\(\displaystyle =\frac{4}{3}\pi-\frac{16}{9}\)

\(D\)を極座標変換すると
\(\displaystyle E=\{(r,\theta)|0\leqq r\leqq \sin\theta,0\leqq \theta\leqq \pi\}\)
よって
\(\displaystyle \iint_D\sqrt{y}dxdy\)
\(\displaystyle =\iint_E\sqrt{r\sin\theta}・rdrd\theta\)
\(\displaystyle =\int_0^\pi\int_0^{\sin\theta}r^\frac{3}{2}\sqrt{\sin\theta}drd\theta\)
\(\displaystyle =\int_0^\pi\frac{2}{5}\sin^3\theta d\theta\)
\(\displaystyle =\frac{4}{5}\int_0^\frac{\pi}{2}\sin^3\theta d\theta\)
\(\displaystyle =\frac{8}{15}\)