AP Calculus Applications Integration Quiz

Contents

This ap calculus applications integration quiz asks 10 multiple-choice questions on applications of integration, and each answer comes with a short explanation.

Ten multiple-choice questions check average value, motion, area between curves, volumes and accumulation. Pick one answer for each question, then read the explanation to see the typical mistake behind each wrong choice.

AP Calculus ABCommon Core: AP Calculus AB Unit 8

🎯 Applications of Integration – AP Calculus AB math quiz

Score: 0 / 10

Answered: 0 / 10

Question 1

What is the average value of \(f(x)=6x\) on \([2,5]\)?

A \(63\)

B \(30\)

C \(21\)

D \(12\)

Question 2

A particle has velocity \(v(t)=2t-6\) for \(0\le t\le 6\). What is the total distance traveled?

A \(0\)

B \(36\)

C \(18\)

D \(9\)

Question 3

The region.

What is the area of the shaded region between \(y=2x\) and \(y=x^{2}\)?

A \(4\)

B \(\frac{8}{3}\)

C \(2\)

D \(\frac{4}{3}\)

Question 4

The area under the parabola \(y=x^{2}\) between \(x=0\) and \(x=2\) spins around the \(x\)-axis. What is the volume of the solid?

A \(\frac{32\pi}{5}\)

B \(\frac{8\pi}{3}\)

C \(\frac{16\pi}{5}\)

D \(32\pi\)

Question 5

The region.

This region is revolved about the \(x\)-axis. Which integral gives the volume?

A \(\pi\int_{0}^{1}\left(x-x^{2}\right)^{2}dx\)

B \(\pi\int_{0}^{1}\left(x-x^{2}\right)dx\)

C \(\pi\int_{0}^{1}\left(x^{2}-x^{4}\right)dx\)

D \(\int_{0}^{1}\left(x^{2}-x^{4}\right)dx\)

Question 6

The base of a solid is the region on \([0,2]\) under the line \(y=x\) and above the \(x\)-axis. Cross sections perpendicular to the \(x\)-axis are squares. What is the volume?

A \(\frac{4}{3}\)

B \(\frac{8}{3}\)

C \(2\)

D \(4\)

Question 7

Amount increases while the rate in exceeds the rate out, and decreases afterward.

The graph shows the rate in \(10-t\) and the rate out \(2+t\) for the amount in a tank. At which time on \(0\le t\le 8\) is the amount greatest?

A \(t=4\)

B \(t=0\)

C \(t=6\)

D \(t=8\)

Question 8

What is the area of the region bounded by \(x=y^{2}\) and \(x=4\)?

A \(\frac{16}{3}\)

B \(\frac{32}{3}\)

C \(16\)

D \(\frac{64}{3}\)

Question 9

Velocity of a particle for \(0\le t\le 4\).

The graph shows the velocity of a particle for \(0\le t\le 4\). What is the total distance traveled?

A \(0\)

B \(16\)

C \(4\)

D \(8\)

Question 10

Water in a tank is \(A(0)=30\) gallons and \(A^{\prime}(t)=6-2t\) gallons per minute. What is \(A(4)\)?

A \(38\)

B \(8\)

C \(22\)

D \(36\)

🎉 Quiz complete!

0 / 10

How does this quiz work?

Read each question, choose one answer among A, B, C and D, then check it: a short explanation appears after every answer. Your score out of 10 is shown at the end, and you can restart the quiz as often as you like.

What should you review before the quiz?

Reread the key facts of the lesson and the worked example. If a question is hard, go back to the matching part of the lesson, then try the quiz again. Write your work on scrap paper: most mistakes come from skipping a step, not from the idea itself.

Remember:

Average value: \(\frac{1}{b-a}\int_{a}^{b}f(x)\,dx\).
Area between curves: integrate top minus bottom, or right minus left.
Volume with cross sections: \(\int A(x)\,dx\); washers: \(\pi\int(R^{2}-r^{2})\,dx\).
Net change: \(A(b)=A(a)+\int_{a}^{b}A^{\prime}(t)\,dt\).