单选题 （共 3 题 ），每题只有一个选项正确
设有空间区域 $\Omega_{1}: x^{2}+y^{2}+z^{2} \leqslant R^{2}, z \geqslant 0$; 及 $\Omega_{2}: x^{2}+y^{2}+z^{2} \leqslant R^{2}, x \geqslant 0, y \geqslant 0, z \geqslant 0$, 则( )
$\text{A.}$ $\iiint_{\Omega_{1}} x \mathrm{~d} v=4 \iiint_{\Omega_{2}} x \mathrm{~d} v$.
$\text{B.}$ $\iiint_{\Omega_{1}} y \mathrm{~d} v=4 \iiint_{\Omega_{2}} y \mathrm{~d} v$.
$\text{C.}$ $\iiint_{\Omega_{1}} z \mathrm{~d} v=4 \iiint_{\Omega_{2}} z \mathrm{~d} v$.
$\text{D.}$ $\iiint_{\Omega_{1}} x y z \mathrm{~d} v=4 \iiint_{\Omega_{2}} x y z \mathrm{~d} v$.
设函数 $f(x, y)=1+\frac{x y}{\sqrt{1+y^3}}$, 则积分 $I=\int_0^1 \mathrm{~d} x \int_{x^2}^1 f(x, y) \mathrm{d} y=$
$\text{A.}$ $\frac{1}{3}(\sqrt{2}+1)$.
$\text{B.}$ $\frac{1}{6}(\sqrt{2}-1)$.
$\text{C.}$ $\frac{1}{6}(\sqrt{2}+1)$.
$\text{D.}$ $\frac{1}{3}(\sqrt{2}-1)$.
设二重积分 $I_1=\iint_D \frac{x+y-1}{4} \mathrm{~d} x \mathrm{~d} y, I_2=\iint_D\left(\frac{x+y-1}{4}\right)^2 \mathrm{~d} x \mathrm{~d} y, I_3=\iint_D\left(\frac{x+y-1}{4}\right)^3 \mathrm{~d} x \mathrm{~d} y$, 其 中 $D=\left\{(x, y) \mid(x-2)^2+(y-1)^2 \leqslant 2\right\}$, 则 $I_1, I_2, I_3$ 的大小关系为
$\text{A.}$ $I_1 < I_2 < I_3$.
$\text{B.}$ $I_3 < I_2 < I_1$.
$\text{C.}$ $I_3 < I_1 < I_2$.
$\text{D.}$ $I_2 < I_3 < I_1$.