AP Calculus AB Average Value, Area and Volume Test with Point Values (PDF)

AP Calculus AB math test: Average Value, Area and Volume – Maths PDF

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This ap calculus area volume test for AP Calculus AB covers average value, area and volume (Common Core AP CALC AB 8.1, AP CALC AB 8.2, AP CALC AB 8.3, AP CALC AB 8.4, AP CALC AB 8.5, AP CALC AB 8.6, AP CALC AB 8.7, AP CALC AB 8.8, AP CALC AB 8.9, AP CALC AB 8.10, AP CALC AB 8.11, AP CALC AB 8.12), with a free PDF to print.

This test checks average value, motion with integrals, area between curves, volumes with cross sections and discs and washers (AP Calculus AB, 8.1 to 8.12). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.

Time and points

About this test

Grade: AP Calculus AB
Chapter: Average Value, Area and Volume
Standards: AP CALC AB 8.1, AP CALC AB 8.2, AP CALC AB 8.3, AP CALC AB 8.4, AP CALC AB 8.5, AP CALC AB 8.6, AP CALC AB 8.7, AP CALC AB 8.8, AP CALC AB 8.9, AP CALC AB 8.10, AP CALC AB 8.11, AP CALC AB 8.12
Time: 45 minutes
Total: 20 points

Ap calculus area volume test: questions

Question 1 – Average value [2 pts]

Find the average value of \(f(x)=e^{2x}\) on \([0,\ln 3]\).

Question 2 – Motion [2 pts]

A bike has velocity \(v(t)=6-2t\) m/s for \(0\le t\le 5\). Find the displacement and the total distance.

Question 3 – Area between curves [3 pts]

The region between y = x + 2 and y = x².

Find the area of the region between \(y=x+2\) and \(y=x^{2}\).

Question 4 – Area with a crossing [3 pts]

Find the total area between \(y=\sin x\) and \(y=\cos x\) for \(0\le x\le\pi\).

Question 5 – Cross sections [3 pts]

Base region under y = √x.

The base of a solid is the region under \(y=\sqrt{x}\) on \([0,4]\). Cross sections perpendicular to the \(x\)-axis are equilateral triangles with a side in the base. Find the volume.

Question 6 – Washer method [3 pts]

Outer radius 2 and inner radius x.

The region bounded by \(y=2\), \(y=x\) and the \(y\)-axis is revolved about the \(x\)-axis. Find the volume.

Question 7 – Average rate in context [2 pts]

A traffic sensor reports cars per minute as \(r(t)=20+12\sin\frac{\pi t}{30}\) for \(0\le t\le 30\) minutes. Find the average rate and the total number of cars.

Question 8 – Volume of a solid of revolution [2 pts]

Find the volume when the region under \(y=e^{x}\) on \([0,1]\) is revolved about the \(x\)-axis.

Key facts behind this ap calculus area volume test

Key facts

Average value = (1/(b – a)) times the integral of f over [a, b].
Displacement is the integral of v; total distance is the integral of |v|.
Area between curves = integral of (upper – lower); split the interval at every crossing.
Volume by cross sections = integral of A(x); disc method = pi times the integral of (radius squared); washers subtract the inner radius squared.

How to use this ap calculus area volume test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All AP Calculus AB chapters: AP Calculus AB math.

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