累次積分(repeated integrals)

演習問題

1.: 次の2重積分を計算しよう．
(a): $\displaystyle{\iint_{\Omega}x^2 dxdy, \Omega: -1 \leq x \leq 1, 0 \leq y \leq 3}$
(b): $\displaystyle{\iint_{\Omega}e^{x+y} dxdy, \Omega: 0 \leq x \leq 1, 0 \leq y \leq x}$
(c): $\displaystyle{\iint_{\Omega}\sqrt{xy} dxdy, \Omega: 0 \leq y \leq 1, y^2 \leq x \leq y}$
(d): $\displaystyle{\iint_{\Omega}(4 - y^2) dxdy, \Omega}$ はとで囲まれた領域
2.: 次の積分順序の交換をしよう．
(a): $\displaystyle{\int_{0}^{1}\int_{x^4}^{x^2}f(x,y)dydx}$
(b): $\displaystyle{\int_{0}^{1}\int_{-y}^{y}f(x,y)dxdy}$
(c): $\displaystyle{\int_{1}^{4}\int_{x}^{2x}f(x,y)dydx}$
3.: 次の2重積分を計算しよう．
(a): $\displaystyle{\int_{0}^{1}\int_{y}^{1}e^{y/x}dxdy}$
(b): $\displaystyle{\int_{0}^{1}\int_{x}^{1}e^{y^2}dydx}$
(c): $\displaystyle{\int_{0}^{1} dy \int_{y}^{\sqrt{y}}\frac{\sin{x}}{x}dx}$