【微分積分】8-4-1 定符号関数の広義重積分｜要点まとめ
このページでは、大学数学の微分積分で扱う「定符号関数の広義重積分」について解説します。定符号関数や近似増加列の基本的な定義を整理し、極座標変換や変数変換を用いた実践的な計算例題11選と丁寧な解答を掲載しています。
広義重積分
【定符号関数】
関数\(f(x,y)\)が\(D\)上で常に\(f(x,y)\geqq0\)または\(D\)上で常に\(f(x,y)\leqq0\)であるとき、\(f(x,y)\)を\(D\)上の定符号関数という。
【近似増加列】
\(D\)を有界とは限らない\(\mathbb{R}^2\)を部分集合とする。\(D\)の部分集合の列\(\{D_n\}_{n=1}^\infty\)が次の3条件を満たすとき、\(D\)の近似増加列という。
(1)\(D_n\)は面積確定な有界閉領域である。
(2)\(D_1\subset D_2\subset \cdots \subset D_n\subset \cdots\)
(3)任意の有界閉集合\(K\subset D\)に対して、\(K\subset D_n\)となる自然数\(n\)が存在する。
【広義重積分】
\(D\)の近似増加列\(D_n\)のとき、
\(\displaystyle \iint_Df(x,y)dxdy:=\lim_{n\to\infty}\iint_{D_n}f(x,y)dxdy\)
と定義される。これを\(f(x,y)\)の\(D\)における広義の2重積分または広義重積分という。この極限値が存在するときに広義重積分は収束するといい、\(\infty\)であるときに広義重積分は発散するという。広義重積分が収束するとき、\(D\)上で広義重積分可能であるという。
【広義積分の定義の妥当性】
\(f(x,y)\)は\(D\)上の連続な定符号関数とする。このとき、\(D\)の2個の近似増加列\(\{D_n\}_{n=1}^\infty,\{E_n\}_{n=1}^\infty\)に対して
\(\displaystyle \lim_{n\to\infty}\iint_{D_n}f(x,y)dxdy=\lim_{n\to\infty}\iint_{E_n}f(x,y)dxdy\)
が成り立つ。
【例題】次の広義重積分を求めなさい。
(1)\(\displaystyle \iint_De^{-x-y}dxdy\)
\(D=\{(x,y)|x\geqq0,y\geqq0\}\)
\(D=\{(x,y)|x\geqq0,y\geqq0\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\{(x,y)|0\leqq x\leqq n,0\leqq y\leqq n\}\)
よって
\(\displaystyle \iint_De^{-x-y}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}e^{-x-y}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_0^ne^{-x}dx\int_0^ne^{-y}dy\)
\(\displaystyle =\lim_{n\to\infty}([-e^{-x}]_0^n)^2\)
\(\displaystyle =\lim_{n\to\infty}(1-e^{-n})^2\)
\(\displaystyle =1\)
\(D_n=\{(x,y)|0\leqq x\leqq n,0\leqq y\leqq n\}\)
よって
\(\displaystyle \iint_De^{-x-y}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}e^{-x-y}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_0^ne^{-x}dx\int_0^ne^{-y}dy\)
\(\displaystyle =\lim_{n\to\infty}([-e^{-x}]_0^n)^2\)
\(\displaystyle =\lim_{n\to\infty}(1-e^{-n})^2\)
\(\displaystyle =1\)
(2)\(\displaystyle \iint_Dx^2e^{-x^2-y^2}dxdy\)
\(D=\{(x,y)|x\geqq0,y\geqq0\}\)
\(D=\{(x,y)|x\geqq0,y\geqq0\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\{(x,y)|x^2+y^2\leqq n^2,x\geqq0,y\geqq0\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta)|0\leqq r\leqq n,0\leqq \theta\leqq \frac{\pi}{2}\right\}\)
よって
\(\displaystyle \iint_Dx^2e^{-x^2-y^2}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{E_n}(r\cos\theta)^2e^{-r^2}・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^nr^3e^{-r^2}dr\int_0^\frac{\pi}{2}\cos^2\theta d\theta\)
\(\displaystyle =\lim_{n\to\infty}\frac{\pi}{4}\int_0^nr^3e^{-r^2}dr\)
\(\displaystyle =\lim_{n\to\infty}\frac{\pi}{4}・\frac{1}{2}\int_0^{n^2}te^{-t}dt\)
\(\displaystyle =\lim_{n\to\infty}\frac{\pi}{8}\left(1-\frac{n^2+1}{e^{n^2}}\right)\)
\(\displaystyle =\frac{\pi}{8}\)
\(D_n=\{(x,y)|x^2+y^2\leqq n^2,x\geqq0,y\geqq0\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta)|0\leqq r\leqq n,0\leqq \theta\leqq \frac{\pi}{2}\right\}\)
よって
\(\displaystyle \iint_Dx^2e^{-x^2-y^2}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{E_n}(r\cos\theta)^2e^{-r^2}・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^nr^3e^{-r^2}dr\int_0^\frac{\pi}{2}\cos^2\theta d\theta\)
\(\displaystyle =\lim_{n\to\infty}\frac{\pi}{4}\int_0^nr^3e^{-r^2}dr\)
\(\displaystyle =\lim_{n\to\infty}\frac{\pi}{4}・\frac{1}{2}\int_0^{n^2}te^{-t}dt\)
\(\displaystyle =\lim_{n\to\infty}\frac{\pi}{8}\left(1-\frac{n^2+1}{e^{n^2}}\right)\)
\(\displaystyle =\frac{\pi}{8}\)
(3)\(\displaystyle \iint_De^{-x^2-y^2}dxdy\)
\(D=\{(x,y)|x\geqq0,y\geqq0\}\)
\(D=\{(x,y)|x\geqq0,y\geqq0\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\{(x,y)|x^2+y^2\leqq n^2,x\geqq0,y\geqq0\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta)|0\leqq r\leqq n,0\leqq \theta\leqq \frac{\pi}{2}\right\}\)
よって
\(\displaystyle \iint_De^{-x^2-y^2}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{E_n}e^{-r^2}・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^nre^{-r^2}dr\int_0^\frac{\pi}{2}d\theta\)
\(\displaystyle =\lim_{n\to\infty}\frac{\pi}{4}(1-e^{-n^2})\)
\(\displaystyle =\frac{\pi}{4}\)
\(D_n=\{(x,y)|x^2+y^2\leqq n^2,x\geqq0,y\geqq0\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta)|0\leqq r\leqq n,0\leqq \theta\leqq \frac{\pi}{2}\right\}\)
よって
\(\displaystyle \iint_De^{-x^2-y^2}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{E_n}e^{-r^2}・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^nre^{-r^2}dr\int_0^\frac{\pi}{2}d\theta\)
\(\displaystyle =\lim_{n\to\infty}\frac{\pi}{4}(1-e^{-n^2})\)
\(\displaystyle =\frac{\pi}{4}\)
(4)\(\displaystyle \iint_D\frac{1}{\sqrt{1-x^2-y^2}}dxdy\)
\(D=\{(x,y)|x^2+y^2< 1\}\)
\(D=\{(x,y)|x^2+y^2< 1\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\left\{(x,y)|x^2+y^2\leqq 1-\frac{1}{n}\right\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta)|0\leqq r\leqq \sqrt{1-\frac{1}{n}},0\leqq \theta\leqq 2\pi\right\}\)
よって
\(\displaystyle \iint_D\frac{1}{\sqrt{1-x^2-y^2}}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{E_n}\frac{1}{\sqrt{1-r^2}}・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^\sqrt{1-\frac{1}{n}}\frac{r}{\sqrt{1-r^2}}dr\int_0^{2\pi}d\theta\)
\(\displaystyle =\lim_{n\to\infty}2\pi\left(1-\frac{1}{\sqrt{n}}\right)\)
\(\displaystyle =2\pi\)
\(D_n=\left\{(x,y)|x^2+y^2\leqq 1-\frac{1}{n}\right\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta)|0\leqq r\leqq \sqrt{1-\frac{1}{n}},0\leqq \theta\leqq 2\pi\right\}\)
よって
\(\displaystyle \iint_D\frac{1}{\sqrt{1-x^2-y^2}}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{E_n}\frac{1}{\sqrt{1-r^2}}・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_0^\sqrt{1-\frac{1}{n}}\frac{r}{\sqrt{1-r^2}}dr\int_0^{2\pi}d\theta\)
\(\displaystyle =\lim_{n\to\infty}2\pi\left(1-\frac{1}{\sqrt{n}}\right)\)
\(\displaystyle =2\pi\)
(5)\(\displaystyle \iint_D\frac{1}{x+y}dxdy\)
\(D=\{(x,y)|0\leqq x\leqq 1,0\leqq y\leqq 1,(x,y)\neq(0,0)\}\)
\(D=\{(x,y)|0\leqq x\leqq 1,0\leqq y\leqq 1,(x,y)\neq(0,0)\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\left\{(x,y)|\frac{1}{n}\leqq x\leqq 1,0\leqq y\leqq x\right\}\)
\(\ \ \ \cup\left\{(x,y)|\frac{1}{n}\leqq y\leqq 1,0\leqq x\leqq y\right\}\)
よって
\(\displaystyle \iint_D\frac{1}{x+y}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}\frac{1}{x+y}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\left(\int_\frac{1}{n}^1\int_0^x\frac{1}{x+y}dydx+\int_\frac{1}{n}^1\int_0^y\frac{1}{x+y}dxdy\right)\)
\(\displaystyle =\lim_{n\to\infty}\left\{\left(1-\frac{1}{n}\right)\log2+\left(1-\frac{1}{n}\right)\log2\right\}\)
\(\displaystyle =\lim_{n\to\infty}2\left(1-\frac{1}{n}\right)\log2\)
\(\displaystyle =2\log2\)
\(D_n=\left\{(x,y)|\frac{1}{n}\leqq x\leqq 1,0\leqq y\leqq x\right\}\)
\(\ \ \ \cup\left\{(x,y)|\frac{1}{n}\leqq y\leqq 1,0\leqq x\leqq y\right\}\)
よって
\(\displaystyle \iint_D\frac{1}{x+y}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}\frac{1}{x+y}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\left(\int_\frac{1}{n}^1\int_0^x\frac{1}{x+y}dydx+\int_\frac{1}{n}^1\int_0^y\frac{1}{x+y}dxdy\right)\)
\(\displaystyle =\lim_{n\to\infty}\left\{\left(1-\frac{1}{n}\right)\log2+\left(1-\frac{1}{n}\right)\log2\right\}\)
\(\displaystyle =\lim_{n\to\infty}2\left(1-\frac{1}{n}\right)\log2\)
\(\displaystyle =2\log2\)
(6)\(\displaystyle \iint_D\log(x^2+y^2)dxdy\)
\(D=\{(x,y)|0< x^2+y^2\leqq 1\}\)
\(D=\{(x,y)|0< x^2+y^2\leqq 1\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\left\{(x,y)|\frac{1}{n^2}\leqq x^2+y^2\leqq 1\right\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta)|\frac{1}{n}\leqq r\leqq 1,0\leqq \theta\leqq 2\pi\right\}\)
よって
\(\displaystyle \iint_D\log(x^2+y^2)dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{E_n}\log r^2・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^12r\log rdr\int_0^{2\pi}d\theta\)
\(\displaystyle =\lim_{n\to\infty}\left\{-\pi+\frac{\pi}{n^2}(1+2\log n)\right\}\)
\(\displaystyle =-\pi\)
\(D_n=\left\{(x,y)|\frac{1}{n^2}\leqq x^2+y^2\leqq 1\right\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta)|\frac{1}{n}\leqq r\leqq 1,0\leqq \theta\leqq 2\pi\right\}\)
よって
\(\displaystyle \iint_D\log(x^2+y^2)dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{E_n}\log r^2・rdrd\theta\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^12r\log rdr\int_0^{2\pi}d\theta\)
\(\displaystyle =\lim_{n\to\infty}\left\{-\pi+\frac{\pi}{n^2}(1+2\log n)\right\}\)
\(\displaystyle =-\pi\)
(7)\(\displaystyle \iint_D\frac{xy}{(x^2+y^2)^3}dxdy\)
\(D=\{(x,y)|x\geqq1,y\geqq1\}\)
\(D=\{(x,y)|x\geqq1,y\geqq1\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\{(x,y)|1\leqq x\leqq n,1\leqq y\leqq n\}\)
よって
\(\displaystyle \iint_D\frac{xy}{(x^2+y^2)^3}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}\frac{xy}{(x^2+y^2)^3}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_1^n\int_1^n \frac{xy}{(x^2+y^2)^3}dydx\)
\(\displaystyle =\lim_{n\to\infty}\frac{1}{4}\int_1^n\left\{\frac{x}{(x^2+1)^2}-\frac{x}{(x^2+n^2)^2}\right\}dx\)
\(\displaystyle =\lim_{n\to\infty}\left\{\frac{1}{16}+\frac{1}{16n^2}-\frac{1}{4(n^2+1)}\right\}\)
\(\displaystyle =\frac{1}{16}\)
\(D_n=\{(x,y)|1\leqq x\leqq n,1\leqq y\leqq n\}\)
よって
\(\displaystyle \iint_D\frac{xy}{(x^2+y^2)^3}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}\frac{xy}{(x^2+y^2)^3}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_1^n\int_1^n \frac{xy}{(x^2+y^2)^3}dydx\)
\(\displaystyle =\lim_{n\to\infty}\frac{1}{4}\int_1^n\left\{\frac{x}{(x^2+1)^2}-\frac{x}{(x^2+n^2)^2}\right\}dx\)
\(\displaystyle =\lim_{n\to\infty}\left\{\frac{1}{16}+\frac{1}{16n^2}-\frac{1}{4(n^2+1)}\right\}\)
\(\displaystyle =\frac{1}{16}\)
(8)\(\displaystyle \iint_De^\frac{y}{x}dxdy\)
\(D=\{(x,y)|0< x\leqq1,0\leqq y\leqq x^2\}\)
\(D=\{(x,y)|0< x\leqq1,0\leqq y\leqq x^2\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\left\{(x,y)|\frac{1}{n}\leqq x\leqq 1,0\leqq y\leqq x^2\right\}\)
よって
\(\displaystyle \iint_De^\frac{y}{x}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}e^\frac{y}{x}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\int_0^{x^2}e^\frac{y}{x}dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1(xe^x-x)dx\)
\(\displaystyle =\lim_{n\to\infty}\left\{\left(1-\frac{1}{n}\right)e^\frac{1}{n}-\frac{1}{2}+\frac{1}{2n^2}\right\}\)
\(\displaystyle =\frac{1}{2}\)
\(D_n=\left\{(x,y)|\frac{1}{n}\leqq x\leqq 1,0\leqq y\leqq x^2\right\}\)
よって
\(\displaystyle \iint_De^\frac{y}{x}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}e^\frac{y}{x}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\int_0^{x^2}e^\frac{y}{x}dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1(xe^x-x)dx\)
\(\displaystyle =\lim_{n\to\infty}\left\{\left(1-\frac{1}{n}\right)e^\frac{1}{n}-\frac{1}{2}+\frac{1}{2n^2}\right\}\)
\(\displaystyle =\frac{1}{2}\)
(9)\(\displaystyle \iint_Dx^2ye^{-y^3}dxdy\)
\(D=\{(x,y)|x\geqq0,y\geqq x^3\}\)
\(D=\{(x,y)|x\geqq0,y\geqq x^3\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\{(x,y)|0\leqq y\leqq n,0\leqq x\leqq \sqrt[3]{y}\}\)
よって
\(\displaystyle \iint_Dx^2ye^{-y^3}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}x^2ye^{-y^3}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_0^n\int_0^\sqrt[3]{y}x^2ye^{-y^3}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_0^n\frac{y^2}{3}e^{-y^3}dy\)
\(\displaystyle =\lim_{n\to\infty}\frac{1}{9}(1-e^{-n^3})\)
\(\displaystyle =\frac{1}{9}\)
\(D_n=\{(x,y)|0\leqq y\leqq n,0\leqq x\leqq \sqrt[3]{y}\}\)
よって
\(\displaystyle \iint_Dx^2ye^{-y^3}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\iint_{D_n}x^2ye^{-y^3}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_0^n\int_0^\sqrt[3]{y}x^2ye^{-y^3}dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_0^n\frac{y^2}{3}e^{-y^3}dy\)
\(\displaystyle =\lim_{n\to\infty}\frac{1}{9}(1-e^{-n^3})\)
\(\displaystyle =\frac{1}{9}\)
(10)\(\displaystyle \iiint_D\frac{1}{(x^2+y^2+z^2)^2}dxdydz\)
\(D=\{(x,y,z)|x^2+y^2+z^2\geqq1\}\)
\(D=\{(x,y,z)|x^2+y^2+z^2\geqq1\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\{(x,y,z)|1\leqq x^2+y^2+z^2\leqq n^2\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta,\varphi)|1\leqq r\leqq n,0\leqq \theta\leqq \pi,0\leqq \varphi\leqq 2\pi\right\}\)
よって
\(\displaystyle \iiint_D\frac{1}{(x^2+y^2+z^2)^2}dxdydz\)
\(\displaystyle =\lim_{n\to\infty}\iiint_{E_n}\frac{1}{r^4}・r^2\sin\theta drd\theta d\varphi\)
\(\displaystyle =\lim_{n\to\infty}\int_1^n\frac{1}{r^2}dr\int_0^\pi\sin\theta d\theta\int_0^{2\pi}d\varphi\)
\(\displaystyle =\lim_{n\to\infty}2\pi・2\left[-\frac{1}{r}\right]_1^n\)
\(\displaystyle =\lim_{n\to\infty}4\pi\left(1-\frac{1}{n}\right)\)
\(\displaystyle =4\pi\)
\(D_n=\{(x,y,z)|1\leqq x^2+y^2+z^2\leqq n^2\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta,\varphi)|1\leqq r\leqq n,0\leqq \theta\leqq \pi,0\leqq \varphi\leqq 2\pi\right\}\)
よって
\(\displaystyle \iiint_D\frac{1}{(x^2+y^2+z^2)^2}dxdydz\)
\(\displaystyle =\lim_{n\to\infty}\iiint_{E_n}\frac{1}{r^4}・r^2\sin\theta drd\theta d\varphi\)
\(\displaystyle =\lim_{n\to\infty}\int_1^n\frac{1}{r^2}dr\int_0^\pi\sin\theta d\theta\int_0^{2\pi}d\varphi\)
\(\displaystyle =\lim_{n\to\infty}2\pi・2\left[-\frac{1}{r}\right]_1^n\)
\(\displaystyle =\lim_{n\to\infty}4\pi\left(1-\frac{1}{n}\right)\)
\(\displaystyle =4\pi\)
(11)\(\displaystyle \iiint_D\frac{1}{x^2+y^2+z^2}dxdydz\)
\(D=\{(x,y,z)|0< x^2+y^2+z^2\leqq 1\}\)
\(D=\{(x,y,z)|0< x^2+y^2+z^2\leqq 1\}\)
\(D\)の近似増加列を\(\{D_n\}_{n=1}^\infty\)とすると
\(D_n=\left\{(x,y,z)|\frac{1}{n^2}\leqq x^2+y^2+z^2\leqq 1\right\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta,\varphi)|\frac{1}{n}\leqq r\leqq 1,0\leqq \theta\leqq \pi,0\leqq \varphi\leqq 2\pi\right\}\)
よって
\(\displaystyle \iiint_D\frac{1}{x^2+y^2+z^2}dxdydz\)
\(\displaystyle =\lim_{n\to\infty}\iiint_{E_n}\frac{1}{r^2}・r^2\sin\theta drd\theta d\varphi\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1dr\int_0^\pi\sin\theta d\theta\int_0^{2\pi}d\varphi\)
\(\displaystyle =\lim_{n\to\infty}2\pi・2\left(1-\frac{1}{n}\right)\)
\(\displaystyle =\lim_{n\to\infty}4\pi\left(1-\frac{1}{n}\right)\)
\(\displaystyle =4\pi\)
\(D_n=\left\{(x,y,z)|\frac{1}{n^2}\leqq x^2+y^2+z^2\leqq 1\right\}\)
\(D_n\)を極座標変換すると
\(\displaystyle E_n=\left\{(r,\theta,\varphi)|\frac{1}{n}\leqq r\leqq 1,0\leqq \theta\leqq \pi,0\leqq \varphi\leqq 2\pi\right\}\)
よって
\(\displaystyle \iiint_D\frac{1}{x^2+y^2+z^2}dxdydz\)
\(\displaystyle =\lim_{n\to\infty}\iiint_{E_n}\frac{1}{r^2}・r^2\sin\theta drd\theta d\varphi\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1dr\int_0^\pi\sin\theta d\theta\int_0^{2\pi}d\varphi\)
\(\displaystyle =\lim_{n\to\infty}2\pi・2\left(1-\frac{1}{n}\right)\)
\(\displaystyle =\lim_{n\to\infty}4\pi\left(1-\frac{1}{n}\right)\)
\(\displaystyle =4\pi\)
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