Q: A tank in the shape of a cylinder with radius $$2$$ meters and height $$5$$ meters is filled with water. How much work is required to pump all the water to the top of the tank? (Assume the density of water is $$1000 \ kg/m^3$ and g = 9.8$$ \ $$m/s^2.)$$

$$\displaystyle 1000 \times 9.8 \times \pi \times 2^2$ \times \$$int_0$$^5 (5-y) \, dy$$

Q: Given the polar curve $$r = 2\cos \theta$$, what is the area enclosed by one loop?

$$\displaystyle \$$int_0$$^{\pi} \frac{1}{2} [2\cos \theta]^2 \, d\theta$$

Question 1

Which of the following integrals can be evaluated using the method of integration by parts?

Accepted Answer

$$\int x e^x \, dx$$

Question 2

A tank in the shape of a cylinder with radius $$2$$ meters and height $$5$$ meters is filled with water. How much work is required to pump all the water to the top of the tank? (Assume the density of water is $$1000 \ kg/m^3$ and g = 9.8$$ \ $$m/s^2.)$$

Accepted Answer

$$\displaystyle 1000 	imes 9.8 	imes \pi 	imes 2^2$ 	imes \$$int_0$$^5 (5-y) \, dy$$

Question 3

Which of the following series converges?

Accepted Answer

$$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2$}$$

Question 4

Find the area enclosed between the curves $$y = x^2$ and y = 4$.

Accepted Answer

$$\displaystyle \int_{-2}^{2} (4 - x^2$) \, dx$$

Question 5

Which of the following is the correct formula for the arc length of the curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\displaystyle \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$$

Question 6

Evaluate the improper integral $$\displaystyle \$$int_1$$^{\infty} \frac{1}{x^p} \, dx$$ for which values of $$p$$ does it converge?

Accepted Answer

$$p \u003e 1$$

Question 7

Which of the following is the Maclaurin series for $$e^x$$?

Accepted Answer

$$\displaystyle \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

Question 8

A curve is given in parametric form by $$x = t^2$, y$$ = $$t^3. $$What is $$\frac{dy}{dx} in $$terms of $$t$$?

Accepted Answer

$$\displaystyle \frac{3t}{2}$$

Question 9

Which of the following integrals can be solved using trigonometric substitution?

Accepted Answer

$$\int \frac{1}{\sqrt{4 - x^2$}} \, dx$$

Question 10

Which of the following is the correct formula for the volume of a solid of revolution generated by rotating $$y = f(x)$$ about the $$x-$$axis from $$x = a to$$ $$x = b$$ using the disk method?

Accepted Answer

$$\displaystyle \pi \int_a^b [f(x)]^2 \, dx$$

Question 11

Which of the following is the correct partial fraction decomposition for $$\frac{3x+5}{(x+1)(x-2)}$$?

Accepted Answer

$$\displaystyle \frac{A}{x+1} + \frac{B}{x-2}$$

Question 12

Which of the following tests can be used to determine the convergence of the series $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$?

Accepted Answer

The $$p-$$series test

Question 13

Given the polar curve $$r = 2\cos 	heta$$, what is the area enclosed by one loop?

Accepted Answer

$$\displaystyle \$$int_0$$^{\pi} \frac{1}{2} [2\cos 	heta]^2 \, d	heta$$

Question 14

Which of the following is the correct formula for the Taylor series of $$f(x)$$ centered at $$a$$?

Accepted Answer

$$\displaystyle \sum_{n=0}^{\infty} \frac{f$$^{(n)}$$(a)}{n!} (x-a)^n$$

Question 15

Which of the following is an example of an alternating series?

Accepted Answer

$$\displaystyle \sum_{n=1}^{\infty} (-1$$)^{n+1}$$ \frac{1}{n}$$

Question 16

A spring with natural length $$0.5 m $$requires a force of $$20 N to $$stretch it to $$0.6 m. $$How much work is required to stretch it from $$0.5 m to$$ $$0.7 m$$?

Accepted Answer

$$\displaystyle \int_{0.5}^{0.7} 200(x-0.5) \, dx$$

MAC 2312 Calculus II Syllabus and Study Guide

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