\begin{aligned} \iint_{D} x y\left[1+x^{2}+y^{2}\right] \mathrm{d} x \mathrm{~d} y &=\int_{0}^{\frac{\pi}{2}} \mathrm{~d} \theta \int_{0}^{\sqrt[4]{2}} r^{3} \sin \theta \cos \theta\left[1+r^{2}\right] \mathrm{d} r \\ &=\int_{0}^{\frac{\pi}{2}} \sin \theta \cos \theta \mathrm{d} \theta \int_{0}^{\sqrt[5]{2}} r^{3}\left[1+r^{2}\right] \mathrm{d} r \\ &=\frac{1}{2}\left(\int_{0}^{1} r^{3} \mathrm{~d} r+\int_{1}^{\sqrt[4]{2}} 2 r^{3} \mathrm{~d} r\right)=\frac{3}{8} . \end{aligned}

$$D_{2}=\left\{(x, y) \mid 1 \leqslant x^{2}+y^{2} \leqslant \sqrt{2}, x \geqslant 0, y \geqslant 0\right\},$$

$$\left[1+x^{2}+y^{2}\right]=2,(x, y) \in D_{2} .$$

\begin{aligned} \iint_{D} x y\left[1+x^{2}+y^{2}\right] \mathrm{d} x \mathrm{~d} y &=\iint_{D_{1}} x y \mathrm{~d} x \mathrm{~d} y+\iint_{D_{2}} 2 x y \mathrm{~d} x \mathrm{~d} y \\ &=\int_{0}^{\frac{\pi}{2}} \mathrm{~d} \theta \int_{0}^{1} r^{3} \sin \theta \cos \theta \mathrm{d} r+\int_{0}^{\frac{\pi}{2}} \mathrm{~d} \theta \int_{1}^{\sqrt[3]{2}} 2 r^{3} \sin \theta \cos \theta \mathrm{d} r \\ &=\frac{1}{8}+\frac{1}{4}=\frac{3}{8} . \end{aligned}
①因本站题量较多，无法仔细核对每一个试题，如果试题有误,请点击 编辑进行更正。
②如果您有更好的解答，可以点击 我要评论进行评论。
③如果您想挑战您的朋友，点击 我要分享 下载题目图片发给好友。