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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