Question 1

Determine the area of the region bounded by the graph of $$f(x) = x^2$ and the x-axis as x$ goes from 0$ to 3$.

Accepted Answer

$$\$$int_0$$^3 x^2$ \, dx = 9$$

Question 2

Find the volume of the solid formed by revolving the region bounded by the graphs of $$y = x^2$ + 1$$, $$y = 0$$, $$x = 0$$, and $$x = 1$$ about the y-axis.

Accepted Answer

$$\$$int_0$$^1 2\pi x (x^2$ + 1) \, dx = \frac{5\pi}{2}$$

Question 3

A force of $$12$$ pounds stretches a spring $$2$$ inches. Find the work done in stretching the spring $$1$$ foot. Be sure to keep your units straight.

Accepted Answer

$$72$$ inch-pounds

Question 4

Evaluate the integral: $$\int \frac{2x-1}{x^2$(x+2)} \, dx$$

Accepted Answer

$$\frac{1}{x} - \frac{1}{x^2$} + \frac{1}{2} \ln|x+2| + C$$

Question 5

Evaluate the integral: $$\int \frac{dx}{4x^2$ + 1}$$

Accepted Answer

$$\frac{1}{2} \arctan(2x) + C$$

Question 6

Write the partial fraction decomposition for $$\frac{-x-29}{x^2$+2x-15}.$$

Accepted Answer

$$\frac{2}{x-3} - \frac{7}{x+5}$$

Question 7

Evaluate the improper integral: $$\$$int_0$$^{\infty} e$$^{-x}$$ \, dx$$

Accepted Answer

$$1$$

Question 8

Find the limit as $$n 	o \infty$$: $$\frac{5n^2$}{n^2$ + 2}$$

Accepted Answer

$$5$$

Question 9

Find the sum: $$\sum_{n=0}^{\infty} 4 \left(\frac{2}{3}\right)^n$$

Accepted Answer

$$12$$

Question 10

Show that the series converges or diverges: $$\sum_{n=0}^{\infty} \frac{2^n}{n!}$$

Accepted Answer

The series converges by the ratio test.

Question 11

Find a power series for $$f(x) = \frac{3}{2x-1}$$ centered at $$0. $$Then find the interval of convergence.

Accepted Answer

$$-3 \sum_{n=0}^{\infty} (2x)^n$$, $$|x| \u003c 0.5$$

Calculus Study Guide: Integrals, Series, Limits, and Applications

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