题目
设D=(x,y)mid-sqrt(1-(y-1)^2)le xlesqrt(4-y^2),0le y,le 2已知iintlimits_(D)(x^3+xy^2)/(( sqrt(x^2)+y^(2)+y^{3))^3}dxdy=aln2+b，其中a，b为有理数，求a-b.<|im_end|>请输入你的答案(如果没有答案为数值形式，如4,0,3,9,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,1
设
$D=\left\{(x,y)\mid-\sqrt{1-(y-1)^{2}}\le x\le\sqrt{4-y^{2}},0\le y,\le 2\right\}$
已知$\iint\limits_{D}\frac{x^{3}+xy^{2}}{( \sqrt{x^{2}+y^{2}+y^{3}})^{3}}dxdy=a\ln2+b$，其中a，b为有理数，求a-b.
<|im_end|>
请输入你的答案(如果没有答案为数值形式，如4,0,3,9,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,1
题目解答
答案
将区域 $D$ 转化为极坐标系，其中 $x = r\cos\theta$，$y = r\sin\theta$，$dx\,dy = r\,dr\,d\theta$。
区域 $D$ 在极坐标下为：
$D = \{(r, \theta) \mid 0 \leq \theta \leq \frac{\pi}{2}, 0 \leq r \leq \min(2, 2\sin\theta)\}$
被积函数变为：
$\frac{r^3\cos\theta}{(r + r^3\sin^3\theta)^3} = \frac{\cos\theta}{(1 + r^2\sin^3\theta)^3}$
积分式为：
$\iint_D \frac{\cos\theta}{(1 + r^2\sin^3\theta)^3} r\,dr\,d\theta$
通过换元 $u = 1 + r^2\sin^3\theta$，计算得：
$\int_0^{\frac{\pi}{2}} \frac{\cos\theta}{\sin^3\theta} \left( \frac{1}{2} - \frac{1}{2(1 + 4\sin^5\theta)^2} \right) d\theta = \frac{1}{2}\ln 2 - \frac{1}{4}$
因此，$a = \frac{1}{2}$，$b = -\frac{1}{4}$，$a - b = \frac{3}{4}$。
答案： $\boxed{\frac{3}{4}}$