変数変換(change of variables)

演習問題

1.

重積分を計算しよう．

(a)

$\displaystyle{\iint_{\Omega}x^2dxdy, \Omega = \{(x,y) : x^2 + y^2 \leq 4\}}$

(b)

$\displaystyle{\iint_{\Omega}\log{(x^2+y^2)}dxdy, \Omega = \{(x,y) : 1 \leq x^2 + y^2 \leq 4\}}$

(c)

$\displaystyle{\iint_{\Omega}e^{(y-x)/(y+x)}dxdy, \Omega = \{(x,y) : x+y \leq 1, x \geq 0, y \geq 0 \}}$

(d)

$\displaystyle{\iint_{\Omega}e^{x^2 + y^2}dxdy, \Omega = \{(x,y) : 1 < x^2 + y^2 < 4 \}}$

(e)

$\displaystyle{\iint_{\Omega}\sqrt{1 - x^2 - y^2}dxdy, \Omega = \{(x,y) : x^2 + y^2 \leq 1 \}}$

(f)

$\displaystyle{\iint_{\Omega}(1 - x - 2y)dxdy, \Omega = \{(x,y) : x \geq 0, y \geq 0, x^2 + y^2 \leq 1 \}}$

に変換して，次の積分を求めよう．

$\displaystyle{\iint_{\Omega}(x^2 + y^2) e^{-x+y} dx dy, \Omega = \{-1 \leq x+y \leq 1, -1 \leq x - y \leq 1 \}}$