Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
These 20 problems cover AP Calculus AB Unit 8: average value, motion with integrals, area between curves, volumes by cross sections, discs and washers, and accumulation in applied contexts. Early problems use one formula; later ones need a sketch and several steps.
The last six problems are multi-step: curves that cross, a vertical axis of revolution, an average temperature, a tent volume and a combined area and volume problem.
Standards: AP Calculus AB Unit 8 · Level: Grade 12 AP Calculus AB
Problem 1 Average value of a polynomial ★★★
Compute the mean height of the graph of f(x) = 4x2 − 3x between x = 2 and x = 5.
Problem 2 Average value of sine ★★★
Find the average value of f(x) = sin x on [0, π].
Problem 3 Position from velocity ★★★
A cart starts at position s(0) = 5 meters with velocity v(t) = 4t − t2 m/s. Find its position at t = 3.
Problem 4 Net change and total distance ★★★
A delivery drone flies along a straight line with velocity v(t) = t2 − 7t + 10 meters per second for 0 ≤ t ≤ 6. Find its net change in position and the full path length.
Problem 5 Area between two parabolas ★★★
How much area lies between the downward parabola y = 8 − x2 and the upward parabola y = x2?
Problem 6 Area integrated with respect to y ★★★
Compute the area of the pocket between the sideways parabola x = y2 and the line x = 6 − y.
Problem 7 Square cross sections ★★★
A sculpture has a footprint between the curves y = 2x and y = x2. Slicing it by planes perpendicular to the x-axis gives square slices whose edge spans the footprint. Find the volume of the sculpture.
Problem 8 Semicircular cross sections ★★★
A footbridge model has a base bounded by the x-axis, the curve y = √x and the line x = 9. Each vertical slice, cut perpendicular to the x-axis, is a semicircle whose diameter lies across the base. Find the volume of the model.
Problem 9 Disc method ★★★
Find the volume when the region under y = √(x + 1) from x = 0 to x = 3 is revolved about the x-axis.
Problem 10 Washer method about the x-axis ★★★
Spin the region trapped between y = √x and y = x2 around the x-axis. Find the volume of the solid produced.
Problem 11 Revolving about a horizontal line ★★★
Rotate the region under y = x2 for 0 ≤ x ≤ 1 around the horizontal line y = −1 and find the volume swept out.
Problem 12 Revolving about the y-axis ★★★
Rotate the area between the parabola y = x2 and the y-axis, below the height y = 4, about the y-axis. Integrate in y to get the volume.
Problem 13 Tank with water flowing in and out ★★★
A cistern starts with 120 gallons. For the next 10 minutes the inflow is 2t + 4 gallons per minute while a drain removes 10 gallons per minute. How much water is in the cistern at t = 10, and when is the amount smallest?
Problem 14 Area with three intersection points ★★★
The graphs of y = x3 and y = 4x cross three times. Add the areas of the two bounded pieces.
Problem 15 Area where curves cross ★★★
Find the area of the region between y = sin x and y = cos x on [0, π/2].
Problem 16 Revolving about a vertical line ★★★
Spin the area under y = √x for 0 ≤ x ≤ 4 around the vertical line x = 4 and compute the volume by integrating in y.
Problem 17 Greenhouse temperature ★★★
The temperature inside a greenhouse, in °F, is G(h) = 58 + 12 sin(πh/12), where h counts hours from 6 p.m. Find the mean temperature over the 12 hours from h = 0 to h = 12.
Problem 18 Volume of a tent ★★★
A tent has a base shaped like the region between the x-axis and y = 9 − x2 (feet). Each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in the base. Find the volume.
Problem 19 Area and volume from one region ★★★
A garden bed (in meters) has the shape of the area under y = 1/x between x = 1 and x = e. Find its area, and the volume of the solid made by spinning the bed around the x-axis.
Problem 20 Mean Value Theorem for integrals ★★★
Find the mean value of f(x) = x2 + 1 over [1, 4], then locate a number c in [1, 4] where f(c) equals that mean.
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