【微分積分】8-1-1 区間上の2重積分｜問題集

\(\displaystyle =\int_0^2xdx\int_0^3y^2dy\)
\(\displaystyle =\left[\frac{x^2}{2}\right]_{0}^{2}\left[\frac{y^3}{3}\right]_{0}^{3}\)
\(=2・9\)
\(=18\)

\(\displaystyle =\int_0^1e^xdx\int_0^\pi\sin ydy\)
\(=[e^x]_{0}^{1}[-\cos y]_{0}^{\pi}\)
\(=2(e-1)\)

\(\displaystyle =\iint_K\frac{1}{(1+x^2)(1+y^2)}dxdy\)
\(\displaystyle =\int_0^1\frac{1}{1+x^2}dx\int_0^\sqrt{3}\frac{1}{1+y^2}dy\)
\(=[\tan^{-1}x]_{0}^{1}[\tan^{-1}y]_{0}^{\sqrt{3}}\)
\(\displaystyle =\frac{\pi}{4}・\frac{\pi}{3}\)
\(\displaystyle =\frac{\pi^2}{12}\)

\(\displaystyle =\int_0^1\left(\int_0^1ye^{xy}dx\right)dy\)
\(\displaystyle =\int_0^1[e^{xy}]_{x=0}^{x=1}dy\)
\(\displaystyle =\int_0^1(e^y-1)dy\)
\(=[e^y-y]_{0}^{1}\)
\(=e-2\)

\(\displaystyle =\int_0^\pi\left(\int_0^1y\cos(xy)dx\right)dy\)
\(\displaystyle =\int_0^\pi[\sin(xy)]_{x=0}^{x=1}dy\)
\(\displaystyle =\int_0^\pi\sin ydy\)
\(=[-\cos y]_{0}^{\pi}\)
\(=2\)

\(\displaystyle =\int_0^1\left(\int_0^1\frac{y^2}{(xy)^2+1}dx\right)dy\)
\(\displaystyle =\int_0^1[y\tan^{-1}(xy)]_{x=0}^{x=1}dy\)
\(\displaystyle =\int_0^1y\tan^{-1}ydy\)
\(\displaystyle =\left[\frac{y^2}{2}\tan^{-1}y\right]_{0}^{1}-\int_0^1\frac{y^2}{2}・\frac{1}{1+y^2}dy\)
\(\displaystyle =\frac{\pi}{8}-\frac{1}{2}\int_0^1\left(1-\frac{1}{1+y^2}\right)dy\)
\(\displaystyle =\frac{\pi}{8}-\frac{1}{2}[y-\tan^{-1}y]_0^1\)
\(\displaystyle =\frac{\pi}{4}-\frac{1}{2}\)