一、题目![题目 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/f68a9e590526998388b0f9b71bd5d3f73dda4ed9764819fe8f36488fa537e9b9499f465fd201d7c117b8901c3ad071915a34a688058a739ebc39835753a8d7cc.svg)
求解二重积分:
$$
\int_{0}^{2} \mathrm{d} x \int_{0}^{\sqrt{2 x-x^{2}}} \sqrt{x^{2}+y^{2}} \mathrm{d} y.
$$
难度评级:
二、解析 ![解析 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/6fff698aa5c66c6c7a143e3d2a00fa8ee7eab76be5360d89eb43a03143848e8cd60377c76bf830c93ec6603be5af661d9c52238834792ea548bf14de10b05ad9.svg)
由题知:
$$
x \in (0,2)
$$
$$
y=\sqrt{2 x-x^{2}} \Rightarrow y^{2}=2 x-x^{2} \Rightarrow
$$
$$
x^{2}+y^{2}=2 x \Rightarrow
$$
$$
y^{2}+y^{2}-2 x=0 \Rightarrow(x-1)^{2}+y^{2}=1
$$
且:
$$
y > 0
$$
于是,我们可以绘制出该积分的积分区域:
进而:
$$
\left\{\begin{array}{l}x=r \cos \theta \\ y=r \sin \theta\end{array} \Rightarrow \right.
$$
$$
\sqrt{x^{2}+y^{2}}=\sqrt{(r \cos \theta)^{2}+(r \sin \theta)^{2}} \Rightarrow
$$
$$
\sqrt{r^{2}\left(\sin ^{2} \theta+\cos ^{2} \theta\right)}=r.
$$
于是:
$$
\int_{0}^{2} \mathrm{d} x \int_{0}^{\sqrt{2 x-x^{2}}} \sqrt{x^{2}+y^{2}} \mathrm{d} y=
$$
$$
\int_{0}^{\frac{\pi}{2}} \mathrm{d} \theta \int_{0}^{2 \cos \theta} r \cdot r \mathrm{d} r =
$$
$$
\int_{0}^{\frac{\pi}{2}}\left(\left.\frac{1}{3} r^{3}\right|_{0} ^{2 \cos \theta}\right) \mathrm{d} \theta =
$$
$$
\frac{8}{3} \int_{0}^{\frac{\pi}{2}} \cos ^{3} \theta \mathrm{d} \theta=\frac{8}{3} \times \frac{2}{3} \times 1=\frac{16}{9}.
$$
高等数学![箭头 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/c19692009799eac2a7eb5b9d73167ae3dd6cad169ea3ccdbeb97491b80e87593cfa7384844ec1720d0fb9cf5f00ac456f249d047b61ce2d90bdd241e042f4d89.svg)
涵盖高等数学基础概念、解题技巧等内容,图文并茂,计算过程清晰严谨。
线性代数![箭头 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/c19692009799eac2a7eb5b9d73167ae3dd6cad169ea3ccdbeb97491b80e87593cfa7384844ec1720d0fb9cf5f00ac456f249d047b61ce2d90bdd241e042f4d89.svg)
以独特的视角解析线性代数,让繁复的知识变得直观明了。
特别专题![箭头 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/c19692009799eac2a7eb5b9d73167ae3dd6cad169ea3ccdbeb97491b80e87593cfa7384844ec1720d0fb9cf5f00ac456f249d047b61ce2d90bdd241e042f4d89.svg)
通过专题的形式对数学知识结构做必要的补充,使所学知识更加连贯坚实。