\begin{aligned} & -\iint_{\Sigma_2} \frac{x \mathrm{~d} y \mathrm{~d} z+y \mathrm{~d} z \mathrm{~d} x+z \mathrm{~d} x \mathrm{~d} y}{\sqrt{\left(x^2+y^2+z^2\right)^3}} \\ = & -\iint_{\Sigma_1} \frac{x \mathrm{~d} y \mathrm{~d} z+y \mathrm{~d} z \mathrm{~d} x+z \mathrm{~d} x \mathrm{~d} y}{\sqrt{\left(x^2+y^2+z^2\right)^3}}-\iint_{\Sigma_2} \frac{x \mathrm{~d} y \mathrm{~d} z+y \mathrm{~d} z \mathrm{~d} x+z \mathrm{~d} x \mathrm{~d} y}{\sqrt{\left(x^2+y^2+z^2\right)^3}} \\ = & -\iint_{\Sigma_1} \frac{x \mathrm{~d} y \mathrm{~d} z+y \mathrm{~d} z \mathrm{~d} x+z \mathrm{~d} x \mathrm{~d} y}{\sqrt{\left(x^2+y^2+z^2\right)^3}}+\iiint_{x^2+y^2+z^2 \leqslant \delta^2} \frac{1+1+1}{\delta^3} \mathrm{~d} x \mathrm{~d} y \mathrm{~d} z \\ = & \iint_{x^2+y^2 \leqslant \frac{7}{8}} \frac{\left(-\frac{1}{2}\right) \mathrm{d} x \mathrm{~d} y}{\sqrt{\left(x^2+y^2+\frac{1}{4}\right)^3}}+4 \pi=-\frac{1}{2} \int_0^{2 \pi} \mathrm{d} \theta \int_0^{\sqrt{\frac{7}{8}}} \frac{r}{\sqrt{\left(r^2+\frac{1}{4}\right)^3}} \mathrm{~d} r+4 \pi \\ = & -\frac{\pi}{2} \int_0^{\sqrt{\frac{7}{8}}} \frac{\mathrm{d}\left(r^2+\frac{1}{4}\right)}{\sqrt{\left(r^2+\frac{1}{4}\right)^3}+4 \pi}=\left.\pi \cdot \frac{1}{\sqrt{r^2+\frac{1}{4}}}\right|_{r=0} ^{r=\sqrt{\frac{7}{8}}}+4 \pi=\pi\left(\frac{1}{\sqrt{\frac{9}{8}}}-\frac{1}{\frac{1}{2}}\right)+4 \pi \\ = & \frac{2 \sqrt{2}}{3} \pi+2 \pi \end{aligned}
①点击 首页查看更多试卷和试题 , 点击查看 本题所在试卷