107 學年度台綜大微積分 A 第 8 題

題目

8. (10 Points) Evaluate the double integral

\iint_D \sin(x+y)\,dA

where

D

is the region bounded by

x+y=\pi

x+y=0

and

x-y=\pi

and

x-y=0

令 $u=x+y$ 、 $v=x-y$ 。區域變為 $0\le u,v\le\pi$ ，且

\left|\frac{\partial(x,y)}{\partial(u,v)}\right|=\frac12.

因此

\iint_D\sin(x+y)\,\mathrm{d}A =\frac12\int_0^\pi\int_0^\pi\sin u\,\mathrm{d}u\,\mathrm{d}v =\boxed{\pi}.