单选题 （共 6 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( )
$\text{A.}$ $\iiint_{\Omega_{1}} x \mathrm{~d} v=4 \iiint_{\Omega_{2}} x \mathrm{~d} v$.
$\text{B.}$ $\iiint_{\Omega_{1}} y \mathrm{~d} v=4 \iiint_{\Omega_{2}} y \mathrm{~d} v$.
$\text{C.}$ $\iiint_{\Omega_{1}} z \mathrm{~d} v=4 \iiint_{\Omega_{2}} z \mathrm{~d} v$.
$\text{D.}$ $\iiint_{\Omega_{1}} x y z \mathrm{~d} v=4 \iiint_{\Omega_{2}} x y z \mathrm{~d} v$.
设区域 $D=\left\{(x, y) \mid x^2+y^2 \leq 4, x \geq 0, y \geq 0\right\} ， f(x)$为 $D$ 上的正值连续函数， $a, b$ 为常数，则
$$
\iint_D \frac{a \sqrt{f(x)}+b \sqrt{f(y)}}{\sqrt{f(x)}+\sqrt{f(y)}} \mathrm{d} \sigma=
$$
$\text{A.}$ $a b \pi$
$\text{B.}$ $\frac{a b}{2} \pi$
$\text{C.}$ $(a+b) \pi$
$\text{D.}$ $\frac{a+b}{2} \pi$
设 $f(x, y)$ 为连续函数，则
$$
\int_0^{\frac{\pi}{4}} \mathrm{~d} \theta \int_0^1 f(r \cos \theta, r \sin \theta) r \mathrm{~d} r \text { 等于 }
$$
$\text{A.}$ $\int_0^{\frac{\sqrt{2}}{2}} \mathrm{~d} x \int_x^{\sqrt{1-x^2}} f(x, y) \mathrm{d} y$
$\text{B.}$ $\int_0^{\frac{\sqrt{2}}{2}} \mathrm{~d} x \int_0^{\sqrt{1-x^2}} f(x, y) \mathrm{d} y$
$\text{C.}$ $\int_0^{\frac{\sqrt{2}}{2}} \mathrm{~d} y \int_y^{\sqrt{1-y^2}} f(x, y) \mathrm{d} x$
$\text{D.}$ $\int_0^{\frac{\sqrt{2}}{2}} \mathrm{~d} y \int_0^{\sqrt{1-y^2}} f(x, y) \mathrm{d} x$
设函数 $f(x, y)$ 连续，则二次积分 $\int_{\frac{\pi}{2}}^\pi \mathrm{d} x \int_{\sin x}^1 f(x, y) \mathrm{d} y$等于
$\text{A.}$ $\int_0^1 \mathrm{~d} y \int_{\pi+\arcsin y}^\pi f(x, y) \mathrm{d} x$
$\text{B.}$ $\int_0^1 \mathrm{~d} y \int_{\pi-\arcsin y}^\pi f(x, y) \mathrm{d} x$
$\text{C.}$ $\int_0^1 \mathrm{~d} y \int_{\frac{\pi}{2}}^{\pi+\arcsin y} f(x, y) \mathrm{d} x$
$\text{D.}$ $\int_0^1 \mathrm{~d} y \int_{\frac{\pi}{2}}^{\pi-\arcsin y} f(x, y) \mathrm{d} x$
设函数 $f(x, y)$ 连续，则二次积分 $\int_{\frac{\pi}{2}}^\pi \mathrm{d} x \int_{\sin x}^1 f(x, y) \mathrm{d} y$等于
$\text{A.}$ $\int_0^1 \mathrm{~d} y \int_{\pi+\arcsin y}^\pi f(x, y) \mathrm{d} x$
$\text{B.}$ $\int_0^1 \mathrm{~d} y \int_{\pi-\arcsin y}^\pi f(x, y) \mathrm{d} x$
$\text{C.}$ $\int_0^1 \mathrm{~d} y \int_{\frac{\pi}{2}}^{\pi+\arcsin y} f(x, y) \mathrm{d} x$
$\text{D.}$ $\int_0^1 \mathrm{~d} y \int_{\frac{\pi}{2}}^{\pi-\arcsin y} f(x, y) \mathrm{d} x$
设函数 $f(x, y)$ 连续，则 $\int_1^2 \mathrm{~d} x \int_x^2 f(x, y) \mathrm{d} y+\int_1^2 \mathrm{~d} y \int_y^{4-y} f(x, y) \mathrm{d} x=$
$\text{A.}$ $\int_1^2 \mathrm{~d} x \int_1^{4-x} f(x, y) \mathrm{d} y$
$\text{B.}$ $\int_1^2 \mathrm{~d} x \int_x^{4-x} f(x, y) \mathrm{d} y$
$\text{C.}$ $\int_1^2 \mathrm{~d} y \int_1^{4-y} f(x, y) \mathrm{d} x$
$\text{D.}$ $\int_1^2 \mathrm{~d} y \int_y^2 f(x, y) \mathrm{d} x$