Question

10．$\underset{∬}{D}$$\frac{y}{{x}^{2}+{y}^{2}}$dσ，D是由y²=x，y=x及y=$\sqrt{3}$围成的区域．

Answer 1

分析 根据二重积分的性质和法则计算即可．

解答 解：由$\left\{\begin{array}{l}{{y}^{2}=x}\\{y=x}\end{array}\right.$解得$\left\{\begin{array}{l}{x=1}\\{y=1}\end{array}\right.$，由$\left\{\begin{array}{l}{{y}^{2}=x}\\{y=\sqrt{3}}\end{array}\right.$，解得$\left\{\begin{array}{l}{x=3}\\{y=\sqrt{3}}\end{array}\right.$，
∴$\underset{∬}{D}$$\frac{y}{{x}^{2}+{y}^{2}}$dσ=${∫}_{1}^{\sqrt{3}}$dy${∫}_{y}^{{y}^{2}}$$\frac{y}{{x}^{2}+{y}^{2}}$dx=$\frac{\sqrt{3}π}{12}$-$\frac{1}{2}$ln2

点评 本题考查了二重积分的性质和计算，属于基础题．