題目

Problem

(10) (10 pts) Let $a>0$ . Evaluate the double integral

\int_{-a}^{a}\int_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}}\cos(x^2+y^2)\,dy\,dx.

解答

積分區域是半徑 $a$ 的圓盤，且被積函數只含 $x^2+y^2$ ，改用極座標最直接：

x=r\cos\theta,\qquad y=r\sin\theta,\qquad \mathrm{d}A=r\,\mathrm{d}r\,\mathrm{d}\theta.

因此

\begin{align*} \iint_{x^2+y^2\le a^2}\cos(x^2+y^2)\,\mathrm{d}A =&\,\int_0^{2\pi}\int_0^a\cos(r^2)r\,\mathrm{d}r\,\mathrm{d}\theta \\[4mm] =&\,2\pi\left.\frac12\sin(r^2)\right|_0^a \\[4mm] =&\,\boxed{\pi\sin(a^2)}. \end{align*}