Question
What is the volume of the solid generated by rotating the region enclosed by the curve $y=\frac{3}{x}$ and the $\mathrm{x}$-axis on the interval $[6, \infty)$ about the $x$-axis?
(A) $\frac{\pi}{6}$
(B) $\frac{3 \pi}{6}$
(C) $3 \pi$
(D) divergent
▶️Answer/Explanation
Ans:B
This integral is an improper integral, so rewrite it as a limit and evaluate.
$\begin{aligned} \int_6^{\infty} \pi\left(\frac{3}{x}\right)^2 d x & =\lim _{b \rightarrow \infty} \int_6^b \pi\left(\frac{3}{x}\right)^2 d x=9 \pi \cdot \lim _{b \rightarrow \infty} \int_6^b\left(\frac{1}{x}\right)^2 d x=\left.9 \pi \cdot \lim _{b \rightarrow \infty}\left(-\frac{1}{x}\right)\right|_6 ^b \\ & =9 \pi \cdot \lim _{b \rightarrow \infty}\left(-\frac{1}{b}+\frac{1}{6}\right)=9 \pi \cdot \frac{1}{6}=\frac{3 \pi}{2}\end{aligned}$
Question
You are given two twice-differentiable functions, $f(x)$ and $q(x)$. The table above gives values for $f(x)$ and $g(x)$ and their first and second derivatives at $x=1$. Find $\lim _{x \rightarrow 1} \frac{2 f(x)-6 g(x)}{4 x^2-4 e^{3(x-1)}}$.
(A) $-3$
(B) 1
(C) 2
(D) nonexistent
▶️Answer/Explanation
Ans:A
Question
$4 \int_{-1}^1(1-|x|) d x$
(A) $=0$
(B) $=\frac{1}{2}$
(C) $=1$
(D) does not exist
▶️Answer/Explanation
Ans:C
The given integral is equivalent to $\int_{-1}^0(1+x) d x+\int_0^1(1-x) d x$.
The figure shows the graph of $f(x)=1-|x|$ on $[-1,1]$.
The area of triangle $P Q R$ is equal to $\int_{-1}^1(1-|x|) d x$.
Question
If $x=2 \sin \theta, 0 \leq \theta \leq \frac{\pi}{2}$, then $\int_0^2 \frac{x^2 d x}{\sqrt{4-x^2}}$ is equivalent to:
(A) $4 \int_0^2 \sin ^2 \theta d \theta$
(B) $\int_0^{\pi / 2} 4 \sin ^2 \theta d \theta$
(C) $\int_0^{\pi / 2} 2 \sin \theta \tan \theta d \theta$
(D) $\int_0^2 \frac{2 \sin ^2 \theta}{\cos \theta} d \theta$
▶️Answer/Explanation
Ans:B
Note that, when $x=2 \sin \theta, x^2=4 \sin ^2 \theta, d x=2 \cos \theta d \theta$, and $\sqrt{4-x^2}=2 \cos \theta$. Also, when $x=0, \theta=0$, and when $x=2, \theta=\frac{\pi}{2}$.
Question
Which of the following series converges conditionally?
I. $\sum_{n=0}^{\infty} \frac{(-1)^n 3^n}{2^n}$
II. $\sum_{n=1}^{n=0} \frac{(-1)^n}{\sqrt[3]{n}}$
III. $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^5}$
(A) I only
(B) II only
(C) I and II only
(D) II and III only
▶️Answer/Explanation
Ans:B
Series I is a divergent geometric series with a common ratio of $-\frac{3}{2}$. Series II passes the Alternating Series Test and thus converges. However, the absolute value of series II is a $p$-series with $p=\frac{1}{3}$. Since $p<1$, that $p$ series diverges. Thus, series II is conditionally convergent. Series III passes the Alternating Series Test and thus converges, and the absolute value of series III is a $p$-series with $p=5$. Since $p>1$, that $p$-series converges. Thus, series III is absolutely convergent.