Question 1

Which of the following is the general antiderivative of $$f(x) = 3x^2$?

Accepted Answer

$$x^3 + C$

Question 2

A car’s velocity (in meters per second) is given by $$v(t) = 4t. $$What is the total distance traveled from $$t = 0 to$$ $$t = 5$$ seconds?

Accepted Answer

$$\$$int_0$$^5 4t \, dt = 50$$ meters

Question 3

Which of the following best describes the meaning of $$\int_a^b f'(x) \, dx$$?

Accepted Answer

The net change in $$f(x)$$ from $$x = a to$$ $$x = b$$

Question 4

Find the specific antiderivative of $$f(x) = 2x$$ given that $$F(1) = 5.$$

Accepted Answer

$$F(x) = x^2$ + 3$$

Question 5

Which substitution would you use to evaluate $$\int 2x \cos(x^2$) \, dx$$?

Accepted Answer

$$u = x^2$

Question 6

The rate at which water flows into a tank is $$r(t) = 5t$$ liters per hour. How much water enters the tank between $$t = 2$$ and $$t = 4$$ hours?

Accepted Answer

$$\$$int_2$$^4 5t \, dt = 30$$ liters

Question 7

Set up the definite integral to find the area between $$y = x^2$ and y = 4$ from x = 0$ to x = 2$.

Accepted Answer

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

Question 8

Evaluate $$\$$int_0$$^1 (3x^2$ + 2x) \, dx.$$

Accepted Answer

$$2$$

Question 9

Which of the following is the correct setup for the area between $$x = y^2$ and x = y$ from y = 0$ to y = 1$?

Accepted Answer

$$\$$int_0$$^1 (y - y^2$) \, dy$$

Question 10

If $$f(x) = \sin(x)$$, what is the general antiderivative?

Accepted Answer

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

Question 11

A medication is administered at a rate of $$r(t) = 10e$$^{-0.5t}$$ mg/hour. What is the total amount given in the first 2 hours?

Accepted Answer

$$\$$int_0$$^2 10e$$^{-0.5t}$$ \, dt$$

Question 12

Which of the following is the correct antiderivative of $$f(x) = \frac{1}{x}$$?

Accepted Answer

$$\ln|x| + C$$

Question 13

Find $$\$$int_0$$^2 (6x - 4) \, dx.$$

Accepted Answer

$$8$$

Question 14

Which substitution would simplify $$\int \frac{1}{\sqrt{1 - x^2$}} \, dx$$?

Accepted Answer

$$x = \sin(u)$$

Question 15

If $$F(x) is an $$antiderivative of $$f(x)$$, which statement is true?

Accepted Answer

$$F'(x) = f(x)$$

Question 16

A population grows at a rate $$r(t) = 100e$$^{0.1t}$$ per year. What is the net change in population from t = 0$ to t = 5$ years?

Accepted Answer

$$\$$int_0$$^5 100e$$^{0.1t}$$ \, dt$$

Question 17

Which of the following is the correct setup for the area between $$y = x$$ and $$y = x^2$ from x = 0$ to x = 1$?

Accepted Answer

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

Question 18

Evaluate $$\$$int_1$$^3 (2x) \, dx.$$

Accepted Answer

$$8$$

Question 19

Which of the following is the correct antiderivative of $$f(x) = e$$^{2x}$$?

Accepted Answer

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

Question 20

A chemical reaction produces a substance at a rate $$r(t) = 7t^2$ grams per minute. How much is produced in the first 3 minutes?

Accepted Answer

$$\$$int_0$$^3 7t^2$ \, dt = 63$$ grams

MAT 151 Exam 2 Study Guide: Integration and Applications

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