Q: Which of the following limits is equal to $1$?

$\lim_{x \to 0} \frac{\sin(x)}{x}$

Question 1

Which of the following integrals correctly represents the area between the curves $y = x^2 + 2x + 1$ and $y = 2x + 1$ from $x = 0$ to $x = 1$?

Accepted Answer

$\int_0^1 [(x^2+2x+1) - (2x+1)] \, dx$

Question 2

Given $f(x) = x^3 - 5x + 4$, what is the area between $f(x)$ and the $x$-axis from $x = 0$ to $x = 2$?

Accepted Answer

$\int_0^2 (x^3 - 5x + 4) \, dx$

Question 3

Which of the following is the correct setup for the area bounded by $f(x) = \cos(x)$ and $g(x) = \sin(x)$ from $x = 0$ to $x = \frac{\pi}{2}$?

Accepted Answer

$\int_0^{\frac{\pi}{2}} [\cos(x) - \sin(x)] \, dx$

Question 4

The volume of the solid formed by revolving the region bounded by $y = x^2$ and $y = 2x$ about the $x$-axis from $x = 0$ to $x = 2$ is given by:

Accepted Answer

$\pi \int_0^2 [(2x)^2 - (x^2)^2] \, dx$

Question 5

Which integral represents the volume of the solid generated by revolving the region bounded by $y = 4 - x^2$ and the $x$-axis about the $x$-axis from $x = -2$ to $x = 2$?

Accepted Answer

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

Question 6

Using the disk method, what is the volume of the solid generated by revolving the region bounded by $y = x^2$ and $y = 0$ from $x = 0$ to $x = 3$ about the $x$-axis?

Accepted Answer

$\pi \int_0^3 (x^2)^2 \, dx$

Question 7

Evaluate $\int x e^{x^2} \, dx$.

Accepted Answer

$\frac{1}{2} e^{x^2} + C$

Question 8

Which of the following is the correct result for $\int 	an^2(x) \, dx$?

Accepted Answer

$	an(x) - x + C$

Question 9

Evaluate $\int \frac{1}{x^2 + 4} \, dx$.

Accepted Answer

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

Question 10

Find $\int e^{3x} \sin(2x) \, dx$.

Accepted Answer

$\frac{e^{3x}}{13} (3 \sin(2x) + 2 \cos(2x)) + C$

Question 11

Evaluate $\lim_{x 	o 0} \frac{\sin(5x)}{x}$.

Accepted Answer

$5$

Question 12

Which of the following is the correct result for $\int \ln(x) \, dx$?

Accepted Answer

$x \ln(x) - x + C$

Question 13

Evaluate $\int x \cos(x) \, dx$ using integration by parts.

Accepted Answer

$x \sin(x) + \cos(x) + C$

Question 14

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

Accepted Answer

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

Question 15

Evaluate $\int_0^1 x e^{x^2} \, dx$.

Accepted Answer

$\frac{1}{2}(e - 1)$

Question 16

Which of the following limits is equal to $1$?

Accepted Answer

$\lim_{x 	o 0} \frac{\sin(x)}{x}$

Question 17

Evaluate $\int x^2 e^{x^3} \, dx$.

Accepted Answer

$\frac{1}{3} e^{x^3} + C$

Question 18

Which of the following is the correct result for $\int \sec^2(x) \, dx$?

Accepted Answer

$	an(x) + C$

Question 19

Evaluate $\int_0^{\pi} \sin(x) \, dx$.

Accepted Answer

$0$

Calculus I: Applications of Integrals, Techniques of Integration, and Limits

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