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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