Question 1

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

Accepted Answer

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

Question 2

Evaluate $$\int x e$$^{2x}$$ \, dx$$ using integration by parts.

Accepted Answer

$$\frac{1}{2}x e$$^{2x}$$ - \frac{1}{4}e$$^{2x}$$ + C$$

Question 3

Which formula gives the average value of a continuous function $$f(x)$$ over the interval $$[a, b]$$?

Accepted Answer

$$\frac{1}{b-a} \int_a^b f(x) \, dx$$

Question 4

Given $$f(x) = 5x$$ over $$[1,3]$$, what is the average value of $$f(x) on $$this interval?

Accepted Answer

$$10$$

Question 5

Find the area between the curve $$f(x) = 9 - x^2$ and the x$-axis over $$[1,2]$$.

Accepted Answer

$$\frac{19}{3}$$

Question 6

A rocket's acceleration is given by $$a(t) = 50 - t (in$$ $$m/s^2). $$What is the net change in velocity between $$t=1$$ and $$t=47$$?

Accepted Answer

$$20 m/s$$

Question 7

What is the average acceleration of the rocket between $$t=1$$ and $$t=47$$?

Accepted Answer

$$6.67 m/s$^2$$

Question 8

Which of the following integrals represents the volume of the solid formed by revolving the area between $$f(x) = 9 - x^2$ and the x$-axis over $$[1,2]$$ about the x$-axis?

Accepted Answer

$$V = \pi \$$int_1$$^2 (9 - x^2$)^2$$ \, dx$

Question 9

Evaluate $$\$$int_1$$^2 \sqrt{7x^2$ + 2} \, dx$$ using substitution.

Accepted Answer

$$\frac{1}{14} \left[ (7x^2$ + 2$$)^{3/2}$$ \right]_1^2$

Question 10

If $$f(x) = 3 - x^2$ and g(x) = x$, what is the area between f(x$$)$$ and g(x$$)$$ from x = -1$ to x = 2$?

Accepted Answer

$$\frac{27}{6}$$

Question 11

Which of the following is the correct setup for finding the area between $$f(x) = 3 - x^2$ and g(x) = x$ over x = -1$ to x = 2$?

Accepted Answer

$$\int_{-1}^2 [(3 - x^2$) - x] \, dx$$

Question 12

Which integration technique is most appropriate for $$\int x e$$^{2x}$$ \, dx$$?

Accepted Answer

Integration by parts

Question 13

For $$f(x) = 5x$$ over $$[1,3]$$, which of the following steps correctly finds the average value?

Accepted Answer

$$\frac{1}{3-1} \$$int_1$$^3 5x \, dx$$

Question 14

If $$a(t) = 50 - t$$, what is the formula for the average acceleration over $$[1,47]$$?

Accepted Answer

$$\frac{1}{47-1} \$$int_1$$^{47} (50 - t) \, dt$$

Question 15

Which of the following is the correct result for $$\$$int_1$$^3 x^2$ \, dx$$?

Accepted Answer

$$\frac{26}{3}$$

Question 16

What is the definite integral $$\$$int_1$$^2 (9 - x^2$) \, dx$$?

Accepted Answer

$$\frac{19}{3}$$

Question 17

Which of the following is the correct formula for the volume of revolution about the $$x-$$axis for $$y = 9 - x^2$ over $$[1,2]$$?

Accepted Answer

$$V = \pi \$$int_1$$^2 (9 - x^2$)^2$$ \, dx$

Business Calculus II: Integration, Applications, and Average Value Study Guide

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