### 2022《高等数学A》下册期末考试模拟试卷

$\text{A.}$ $\pi R^2$; $\text{B.}$ $\pi R^3$; $\text{C.}$ $2 \pi R^2$; $\text{D.}$ $2 \pi R^3$.

$\text{A.}$ $\int_L(P+2 x Q) \mathrm{d} s$; $\text{B.}$ $\int_L(2 x P+Q) \mathrm{d} s$; $\text{C.}$ $\int_L \frac{P+2 x Q}{\sqrt{1+4 x^2}} \mathrm{~d} s$; $\text{D.}$ $\int_L \frac{2 x P+Q}{\sqrt{1+4 x^2}} \mathrm{~d} s$.

$\text{A.}$ $\frac{3}{2}$ $\text{B.}$ $\frac{2}{3}$ $\text{C.}$ $\frac{1}{3}$ $\text{D.}$ 2

$\text{A.}$ $\iint_{\Sigma} x \mathrm{~d} S=4 \iint_{\Sigma_1} x \mathrm{~d} S$ $\text{B.}$ $\iint_{\Sigma} y \mathrm{~d} S=4 \iint_{\Sigma_1} y \mathrm{~d} S$ $\text{C.}$ $\iint_{\Sigma} z \mathrm{~d} S=4 \iint_{\Sigma_1} z \mathrm{~d} S$ $\text{D.}$ $\iint_{\Sigma} x y z \mathrm{~d} S=4 \iint_{\Sigma_1} x y z \mathrm{~d} S$

$\text{A.}$ $-\iint_{D_{x y}} x^2 y^2\left(-\sqrt{R^2-x^2-y^2}\right) \mathrm{d} x \mathrm{~d} y, \quad D_{x y}: x^2+y^2 \leq R^2$ $\text{B.}$ $-\iint_{D_{x y}} x^2 y^2 \sqrt{R^2-x^2-y^2} \mathrm{~d} x \mathrm{~d} y$, $D_{x y}: x^2+y^2 \leq R^2$ $\text{C.}$ $\iint_{D_{x y}} x^2 y^2\left(R^2-x^2-y^2\right) \mathrm{d} x \mathrm{~d} y$, $D_{x y}: x^2+y^2 \leq R^2$ $\text{D.}$ $-\iint_{D_{x y}} x^2 y^2\left(R^2-x^2-y^2\right) \mathrm{d} x \mathrm{~d} y$, $D_{x y}: x^2+y^2 \leq R^2$

$\text{A.}$ $\sum_{n=1}^{\infty} \frac{1}{n+n^2}$ $\text{B.}$ $\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right)$ $\text{C.}$ $\sum_{n=1}^{\infty} \sin \frac{1}{\sqrt{n}}$ $\text{D.}$ $\sum_{n=1}^{\infty} \frac{1}{n \sqrt{n}}$

$\text{A.}$ $\sum_{n=1}^{\infty} \frac{\sin n^2}{n^2}$ $\text{B.}$ $\sum_{n=1}^{\infty}(-1)^n \frac{1}{\sqrt{n}}$ $\text{C.}$ $\sum_{n=1}^{\infty}(-1)^n \frac{1}{n}$ $\text{D.}$ $\sum_{n=1}^{\infty}(-1)^n \cdot \frac{n}{n+1}$

$\text{A.}$ $R=\max \left(R_1, R_2\right)$ $\text{B.}$ $R=\min \left(R_1, R_2\right)$ $\text{C.}$ $R=R_1 R_2$ $\text{D.}$ $R=R_1+R_2$

$\text{A.}$ $x+\frac{1}{2} x^2+\frac{1}{3} x^3+\frac{1}{4} x^4+\cdots+\frac{1}{n} x^n+\cdots$ $\text{B.}$ $x-\frac{1}{2} x^2+\frac{1}{3} x^3-\frac{1}{4} x^4+\cdots+(-1)^{n-1} \cdot \frac{1}{n} x^n+\cdots$ $\text{C.}$ $x+\frac{1}{2 !} x^2+\frac{1}{3 !} x^3+\frac{1}{4 !} x^4+\cdots+\frac{1}{n !} x^n+\cdots$ $\text{D.}$ $x-\frac{1}{3 !} x^3+\frac{1}{5 !} x^5-\frac{1}{7 !} x^7+\cdots+(-1)^{n-1} \frac{1}{(2 n-1) !} x^{2 n-1}+\cdots$

$\text{A.}$ $\frac{1}{4}$ $\text{B.}$ $\frac{3}{4}$ $\text{C.}$ $\frac{5}{4}$ $\text{D.}$ $\frac{7}{4}$

$\int_L \mathrm{e}^x(1-2 \cos y) \mathrm{d} x+2 \mathrm{e}^x \sin y \mathrm{~d} y=$
(其中 $L$ 是 $y=\sin x$ 上从点 $A(\pi, 0)$ 到点 $O(0,0)$ 的一段弧 $)$.

(求出满足条件的任何一个函数均可)

$f(x)=\left\{\begin{array}{rr}x & 0 \leq x \leq \pi \\ 0 & -\pi < x < 0\end{array}\right.$, 则 $s(9 \pi)=$

$$L:\left\{\begin{array}{c} x^2+y^2+z^2=\frac{9}{2}, \\ x+z=1 . \end{array}\right.$$

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