单选题 （共 6 题 ），每题只有一个选项正确
1.设 $D=\left\{(x,y)|x²+y²≤4,x≥0,y≥0\right\}$, 且$a>0$,$b>0$,则$ I= \iint _{D} \dfrac {ae^{x^{2}}+be^{y^{2}}}{e^{x^{2}}+e^{y^{2}}}d \sigma =\underline{\quad\quad\quad}$.
$\text{A.}$ $\frac {(a+b)}{4} \pi$
$\text{B.}$ $\frac {(a+b)}{3} \pi$
$\text{C.}$ $\frac {(a+b)}{2} \pi$
$\text{D.}$ $(a+b)\pi$
设 $D=\left\{(x,y)|x+y≤1,x≥0,y≥0\right\}$,令 $I= \iint _{D} \sqrt {x^{2}+y^{2}}dxdy$,$J= \iint _{D} \ln (1+x^{2}+y^{2})dxdy$,$K= \iint _{D}(x^{2}+y^{2})dxdy$, 则
$\text{A.}$ $I < J < K$
$\text{B.}$ $J < K < I$
$\text{C.}$ $J < I < K$
$\text{D.}$ $K < J < I$
$\int _{ \dfrac { \pi }{4}}^{ \dfrac { \pi }{2}}d \theta \int _{0}^{ \cos \theta }r^{2} \cos \theta f(r)dr= \underline{\quad\quad\quad}$.
$\text{A.}$ $\int _{0}^{1}xdx \int _{x}^{1}f(x,y)dy$
$\text{B.}$ $\int _{0}^{1}xdx \int _{0}^{x}f(x,y)y$
$\text{C.}$ $ \int _{0}^{1}xdx \int _{0}^{x}f( \sqrt {x^{2}+y^{2}})dy$
$\text{D.}$ $ \int _{0}^{1}xdx \int _{x}^{1}f( \sqrt {x^{2}+y^{2}})dy$
设$D$是由$A(π,π)$、$B(-π,π)$、$C(-π,-π)$三点构成的三角形区域,则
$\iint _{D}(xy+ \sin x \cos y)dxdy= \underline { \quad \quad \quad }$.
$\text{A.}$ $-π$
$\text{B.}$ $π$
$\text{C.}$ $\dfrac { \pi }{2}$
$\text{D.}$ $0$
设 $D=\left\{(x, y) \mid x^2+y^2 \leq R^2\right\}$, 则 $\iint_D \sqrt{x^2+y^2} \mathrm{~d} \sigma=$.
$\text{A.}$ $\pi R^3$
$\text{B.}$ $\frac{2 \pi R^3}{3}$
$\text{C.}$ $\pi R^2$
$\text{D.}$ $2 \pi R^2$
设函数 $f(x)=\iint_{u^2+v^2 \leqslant x^2} \arctan \left(1+\sqrt{u^2+v^2}\right) \mathrm{d} u \mathrm{~d} v(x>0)$, 则 $\lim _{x \rightarrow 0^{+}} \frac{f(x)}{\mathrm{e}^{-2 x}-1+2 x}=$
$\text{A.}$ $-\frac{\pi^2}{8}$.
$\text{B.}$ $-\frac{\pi^2}{4}$.
$\text{C.}$ $\frac{\pi^2}{4}$.
$\text{D.}$ $\frac{\pi^2}{8}$.