【微分積分】8-4-1 定符号関数の広義重積分｜問題集

\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\left\{(x,y)|\frac{1}{n}\leqq x\leqq 1,0\leqq y\leqq x\right\}\)
よって
\(\displaystyle \iint_De^\frac{y}{x}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}e^\frac{y}{x}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\int_0^xe^\frac{y}{x}dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1x(e-1)dx\)
\(\displaystyle =\lim_{n\to\infty}\left(\frac{1}{2}-\frac{1}{2n^2}\right)\)
\(\displaystyle =\frac{e-1}{2}\)

\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\left\{(x,y)|0\leqq x\leqq 1-\frac{1}{n},0\leqq y\leqq 1-\frac{1}{n}-x\right\}\)
よって
\(\displaystyle \iint_D\frac{x}{\sqrt{1-x-y}}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}\frac{x}{\sqrt{1-x-y}}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_0^{1-\frac{1}{n}}\int_0^{1-\frac{1}{n}-x}\frac{x}{\sqrt{1-x-y}}dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_0^{1-\frac{1}{n}}\left\{-2x\left(\frac{1}{n}\right)^\frac{1}{2}+2x(1-x)^\frac{1}{2}\right\}dx\)
\(\displaystyle =\lim_{n\to\infty}\left\{-\frac{1}{\sqrt{n}}\left(1-\frac{1}{n}\right)^2+\frac{4}{5}\left(\frac{1}{n}\right)^\frac{5}{2}-\frac{4}{3}\left(\frac{1}{n}\right)^\frac{3}{2}-\frac{4}{5}+\frac{4}{3}\right\}\)
\(\displaystyle =\frac{8}{15}\)

\(D\)を極座標変換すると
\(\displaystyle D_n=\left\{(r,\theta)|\frac{1}{n}\leqq r\leqq 1,0\leqq \theta\leqq \frac{\pi}{2}\right\}\)
よって
\(\displaystyle \iint_D\log(x^2+y^2)dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_0^\frac{\pi}{2}\int_\frac{1}{n}^1\log r^2・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^\frac{\pi}{2}\int_\frac{1}{n^2}^1\log t・\frac{dt}{2}d\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^\frac{\pi}{2}\frac{1}{2}\left(-1-\frac{1}{n^2}\log\frac{1}{n^2}+\frac{1}{n^2}\right)d\theta\)
\(\displaystyle =\lim_{n\to\infty}\frac{\pi}{4}\left(-1+\frac{2\log n}{n^2}+\frac{1}{n^2}\right)\)
\(\displaystyle =-\frac{\pi}{4}\)

\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(\displaystyle D_n=\left\{(x,y)|0\leqq x\leqq \left(1-\frac{1}{n}\right)y,\frac{1}{n}\leqq y\leqq 1\right\}\)
よって
\(\displaystyle \iint_D\frac{1}{(y^2-x^2)^\frac{1}{2}}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}\frac{1}{(y^2-x^2)^\frac{1}{2}}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\int_0^{(1-\frac{1}{n})y}\frac{1}{\sqrt{y^2-x^2}}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\sin^{-1}\left(1-\frac{1}{n}\right)dy\)
\(\displaystyle =\frac{\pi}{2}\)

\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(\displaystyle D_n=\left\{(x,y)|0\leqq x\leqq n,0\leqq y\leqq n\right\}\)
よって
\(\displaystyle \iint_De^{-(x+y)}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}e^{-(x+y)}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_0^n\int_0^ne^{-(x+y)}dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_0^n(-e^{-(x+n)}+e^{-x})dx\)
\(\displaystyle =\lim_{n\to\infty}(e^{-2n}-2e^{-n}+1)\)
\(\displaystyle =1\)

\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(\displaystyle D_n=\left\{(x,y)|\frac{2}{n^2}\leqq x^2+y^2\leqq 1,x\geqq0,y\geqq0\right\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta)|\frac{\sqrt{2}}{n}\leqq r\leqq 1,0\leqq \theta\leqq \frac{\pi}{2}\right\}\)
よって
\(\displaystyle \iint_D\tan^{-1}\left(\frac{y}{x}\right)dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{E_n}\tan^{-1}\left(\frac{r\sin\theta}{r\cos\theta}\right)rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^\frac{\pi}{2}\int_\frac{\sqrt{2}}{n}^1\tan^{-1}(\tan\theta)・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^\frac{\pi}{2}\int_\frac{\sqrt{2}}{n}^1\theta rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^\frac{\pi}{2}\theta d\theta\int_\frac{\sqrt{2}}{n}^1rdr\)
\(\displaystyle =\lim_{n\to\infty}\left[\frac{\theta^2}{2}\right]_0^\frac{\pi}{2}\left[\frac{r^2}{2}\right]_\frac{\sqrt{2}}{n}^1\)
\(\displaystyle =\lim_{n\to\infty}\frac{\pi^2}{8}\left(\frac{1}{2}-\frac{1}{n^2}\right)\)
\(\displaystyle =\frac{\pi^2}{16}\)

\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(\displaystyle D_n=\left\{(x,y)|1\leqq x\leqq n,1\leqq y\leqq n\right\}\)
よって
\(\displaystyle \iint_D\frac{1}{x^2y^2}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}\frac{1}{x^2y^2}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_1^n\int_1^n\frac{1}{x^2y^2}dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_1^n\left(\frac{1}{x^2}-\frac{1}{x^2n}\right)dx\)
\(\displaystyle =\lim_{n\to\infty}\left(1-\frac{1}{n^2}\right)\)
\(\displaystyle =1\)

\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(\displaystyle D_n=\left\{(x,y)|y^2+\frac{1}{n}\leqq x\leqq 1,-\sqrt{1-\frac{1}{n}}\leqq y\leqq \sqrt{1-\frac{1}{n}}\right\}\)
よって
\(\displaystyle \iint_D\frac{1}{\sqrt{x-y^2}}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}\frac{1}{\sqrt{x-y^2}}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_{-\sqrt{1-\frac{1}{n}}}^{\sqrt{1-\frac{1}{n}}}\int_{y^2+\frac{1}{n}}^1\frac{1}{\sqrt{x-y^2}}dxdy\)
\(\displaystyle =2\lim_{n\to\infty}\int_{-\sqrt{1-\frac{1}{n}}}^{\sqrt{1-\frac{1}{n}}}\left(\sqrt{1-y^2}-\sqrt{\frac{1}{n}}\right)dy\)
\(\displaystyle =2\lim_{n\to\infty}\left\{\sin^{-1}\sqrt{1-\frac{1}{n}}-\sin^{-1}\left(-\sqrt{1-\frac{1}{n}}\right)\right\}\)
\(\displaystyle \ \ \ -\sqrt{\frac{1}{n}}\left(\sqrt{1-\frac{1}{n}}+\sqrt{1-\frac{1}{n}}\right)\)
\(\displaystyle =\pi\)

\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(\displaystyle D_n=\{(x,y)|x^2+y^2\leqq n^2\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta)|0\leqq r\leqq n,0\leqq \theta\leqq 2\pi\right\}\)
よって
\(\displaystyle \iint_D\frac{1}{1+(x^2+y^2)^2}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{E_n}\frac{1}{1+r^4}・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^{2\pi}\int_0^n\frac{r}{1+r^4}drd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^{2\pi}d\theta\left[\frac{1}{2}\tan^{-1}r^2\right]_0^n\)
\(\displaystyle =\lim_{n\to\infty}2\pi・\frac{1}{2}(\tan^{-1}n^2-\tan^{-1}0)\)
\(\displaystyle =\frac{\pi^2}{2}\)