【微分積分】8-4-2 不定符号関数の広義重積分｜問題集
1.\(\mathbb{R}^2\)の部分集合\(D\)と\(D\)上の関数\(f(x,y)\)を次で定める。
\(\displaystyle f(x,y)=\frac{1}{x}-\frac{1}{y},D=\{(x,y)|0< x\leqq 1,0< y\leqq 1\}\)
(1)\(\displaystyle D_n=\left\{(x,y)|\frac{1}{n}\leqq x\leqq 1,\frac{1}{n}\leqq y\leqq 1\right\}\)に対して、\(\displaystyle \lim_{n\to\infty}\iint_{D_n}f(x,y)dxdy\)を求めなさい。
\(\displaystyle \lim_{n\to\infty}\iint_{D_n}f(x,y)dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\int_\frac{1}{n}^1\left(\frac{1}{x}-\frac{1}{y}\right)dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\left\{\left(1-\frac{1}{n}\right)\frac{1}{x}-\log n\right\}dx\)
\(\displaystyle =\lim_{n\to\infty}\left[\left(1-\frac{1}{n}\right)\log x-x\log n\right]_\frac{1}{n}^1\)
\(\displaystyle =0\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\int_\frac{1}{n}^1\left(\frac{1}{x}-\frac{1}{y}\right)dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\left\{\left(1-\frac{1}{n}\right)\frac{1}{x}-\log n\right\}dx\)
\(\displaystyle =\lim_{n\to\infty}\left[\left(1-\frac{1}{n}\right)\log x-x\log n\right]_\frac{1}{n}^1\)
\(\displaystyle =0\)
(2)\(\displaystyle E_n=\left\{(x,y)|\frac{1}{n^2}\leqq x\leqq 1,\frac{1}{n}\leqq y\leqq 1\right\}\)に対して、\(\displaystyle \lim_{n\to\infty}\iint_{E_n}f(x,y)dxdy\)を求めなさい。
\(\displaystyle \lim_{n\to\infty}\iint_{E_n}f(x,y)dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n^2}^1\int_\frac{1}{n}^1\left(\frac{1}{x}-\frac{1}{y}\right)dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n^2}^1\left\{\left(1-\frac{1}{n}\right)\frac{1}{x}-\log n\right\}dx\)
\(\displaystyle =\lim_{n\to\infty}\left[\left(1-\frac{1}{n}\right)\log x-x\log n\right]_\frac{1}{n^2}^1\)
\(\displaystyle =\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^2\log n\)
\(\displaystyle =\infty\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n^2}^1\int_\frac{1}{n}^1\left(\frac{1}{x}-\frac{1}{y}\right)dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n^2}^1\left\{\left(1-\frac{1}{n}\right)\frac{1}{x}-\log n\right\}dx\)
\(\displaystyle =\lim_{n\to\infty}\left[\left(1-\frac{1}{n}\right)\log x-x\log n\right]_\frac{1}{n^2}^1\)
\(\displaystyle =\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^2\log n\)
\(\displaystyle =\infty\)
(3)広義重積分\(\displaystyle \iint_Df(x,y)dxdy\)の収束・発散を調べなさい。
\(D\)の異なる近似増加列\(\{D_n\}_{n=1}^\infty\)と\(\{E_n\}_{n=1}^\infty\)に対して
\(\displaystyle \lim_{n\to\infty}\iint_{D_n}f(x,y)dxdy\neq\lim_{n\to\infty}\iint_{E_n}f(x,y)dxdy\)
なので、\(\displaystyle \iint_Df(x,y)dxdy\)は発散する。
\(\displaystyle \lim_{n\to\infty}\iint_{D_n}f(x,y)dxdy\neq\lim_{n\to\infty}\iint_{E_n}f(x,y)dxdy\)
なので、\(\displaystyle \iint_Df(x,y)dxdy\)は発散する。
2.\(\mathbb{R}^2\)の部分集合\(D\)と\(D\)上の関数\(f(x,y)\)を次で定める。
\(\displaystyle f(x,y)=\frac{x-y}{(x+y)^3},\)
\(D=\{(x,y)|0\leqq x\leqq 1,0\leqq y\leqq 1,(x,y)\neq(0,0)\}\)
(1)\(\displaystyle \int_0^1\int_0^1f(x,y)dydx\)の値を求めなさい。
\(\displaystyle \int_0^1\int_0^1f(x,y)dydx\)
\(\displaystyle =\lim_{n\to+0}\int_n^1\int_0^1f(x,y)dydx\)
\(\displaystyle =\lim_{n\to+0}\int_n^1\int_0^1\frac{x-y}{(x+y)^3}dydx\)
\(\displaystyle =\lim_{n\to+0}\int_0^1\frac{1}{(x+1)^2}dx\)
\(\displaystyle =\frac{1}{2}\)
\(\displaystyle =\lim_{n\to+0}\int_n^1\int_0^1f(x,y)dydx\)
\(\displaystyle =\lim_{n\to+0}\int_n^1\int_0^1\frac{x-y}{(x+y)^3}dydx\)
\(\displaystyle =\lim_{n\to+0}\int_0^1\frac{1}{(x+1)^2}dx\)
\(\displaystyle =\frac{1}{2}\)
(2)\(\displaystyle \int_0^1\int_0^1f(x,y)dxdy\)の値を求めなさい。
\(\displaystyle \int_0^1\int_0^1f(x,y)dxdy\)
\(\displaystyle =\lim_{n\to+0}\int_n^1\int_0^1f(x,y)dxdy\)
\(\displaystyle =\lim_{n\to+0}\int_n^1\int_0^1\frac{x-y}{(x+y)^3}dxdy\)
\(\displaystyle =\lim_{n\to+0}\int_0^1-\frac{1}{(x+1)^2}dy\)
\(\displaystyle =-\frac{1}{2}\)
\(\displaystyle =\lim_{n\to+0}\int_n^1\int_0^1f(x,y)dxdy\)
\(\displaystyle =\lim_{n\to+0}\int_n^1\int_0^1\frac{x-y}{(x+y)^3}dxdy\)
\(\displaystyle =\lim_{n\to+0}\int_0^1-\frac{1}{(x+1)^2}dy\)
\(\displaystyle =-\frac{1}{2}\)
(3)広義重積分\(\displaystyle \iint_Df(x,y)dxdy\)の収束・発散を調べなさい。
\(f(x,y)\geqq0\)となるのは、
\((x,y)\in D^+=\{(x,y)|0\leqq y\leqq x\leqq 1,(x,y)\neq(0,0)\}\)
\(D^+\)の近似増加列を\(\{D_n^+\}_{n=1}^\infty\)とすると
\(\displaystyle D_n^+=\left\{(x,y)|\frac{1}{n}\leqq x\leqq 1,0\leqq y\leqq x\right\}\)
すなわち
\(\displaystyle \iint_D_n^+f(x,y)dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\int_0^x\frac{x-y}{(x+y)^3}dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\frac{1}{4x}dx\)
\(\displaystyle =\lim_{n\to\infty}\frac{1}{4}\log n\)
\(\displaystyle =\infty\)
よって、\(\displaystyle \iint_Df(x,y)dxdy\)は発散する。
\((x,y)\in D^+=\{(x,y)|0\leqq y\leqq x\leqq 1,(x,y)\neq(0,0)\}\)
\(D^+\)の近似増加列を\(\{D_n^+\}_{n=1}^\infty\)とすると
\(\displaystyle D_n^+=\left\{(x,y)|\frac{1}{n}\leqq x\leqq 1,0\leqq y\leqq x\right\}\)
すなわち
\(\displaystyle \iint_D_n^+f(x,y)dxdy\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\int_0^x\frac{x-y}{(x+y)^3}dydx\)
\(\displaystyle =\lim_{n\to\infty}\int_\frac{1}{n}^1\frac{1}{4x}dx\)
\(\displaystyle =\lim_{n\to\infty}\frac{1}{4}\log n\)
\(\displaystyle =\infty\)
よって、\(\displaystyle \iint_Df(x,y)dxdy\)は発散する。
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