Question

这第4题怎么做？

Answer 1

好像答案和你的不一样-_-||

Answer 2

做坐标变换x=r\cos t，y=r\sin t
则0<=t<=2\pi，0<=r<=\sqrt{R}

追问

能稍微详尽一些么

追答

\iint\limits_D\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)dxdy
=\int_0^{2\pi}dt\int_0^{\sqrt{R}}\left(\frac{1}{a^2}r^2\cos^2t+\frac{1}{b^2}r^2\sin^2t\right)rdr
=\int_0^{2\pi}\left(\frac{1}{a^2}\frac{1+\cos 2t}{2}+\frac{1}{b^2}\frac{1-\cos 2t}{2}\right)\int_0^{\sqrt{R}}r^3dr
=\pi\left(\frac{1}{a^2}}+\frac{1}{b^2}\right)\frac{R^2}{4}
=\frac{\pi}{4}R^2\left(\frac{1}{a^2}}+\frac{1}{b^2}\right)