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AP Calculus BC Practice Quiz: Volumes with Cross Sections: Squares and Rectangles

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

A solid has a known cross-sectional area given by the function A(x) for x in the interval [a, b]. According to the provided content, which mathematical tool is used to calculate the total volume of this solid?

All Questions (7)

A solid has a known cross-sectional area given by the function A(x) for x in the interval [a, b]. According to the provided content, which mathematical tool is used to calculate the total volume of this solid?

A) The derivative of A(x)

B) A definite integral of A(x)

C) The average value of A(x)

D) A Riemann sum with a finite number of rectangles

Correct Answer: B

The provided content explicitly states that we 'Calculate volumes of solids with known cross sections using definite integrals.' The definite integral sums the areas of the infinite, infinitesimally thin cross-sections to find the total volume.

A solid is constructed on a base region in the xy-plane. The cross-sections perpendicular to the x-axis are squares. If the side length of a square cross-section at a given x is represented by s(x), what is the integrand (the function being integrated) used to find the volume of the solid?

A) s(x)

B) [s(x)]^2

C) 2s(x)

D) 4s(x)

Correct Answer: B

The content states that volume is found using definite integrals and the area formulas for the shapes. The area of a square is the side length squared. Therefore, the area of a cross-section at x is A(x) = [s(x)]^2, which serves as the integrand.

A solid has a base defined by the region between two functions, y = f(x) and y = g(x), on the interval [a, b], where f(x) ≥ g(x). The cross-sections perpendicular to the x-axis are rectangles whose height is twice their base. Which definite integral represents the volume of this solid?

A) ∫[a,b] 2(f(x) - g(x)) dx

B) ∫[a,b] (f(x) - g(x))^2 dx

C) ∫[a,b] 2(f(x) - g(x))^2 dx

D) ∫[a,b] (f(x) - g(x) + 2(f(x) - g(x))) dx

Correct Answer: C

The base of the rectangular cross-section is the distance between the curves, which is (f(x) - g(x)). The height is given as twice the base, so height = 2(f(x) - g(x)). The area of the rectangle is base × height = (f(x) - g(x)) * 2(f(x) - g(x)) = 2(f(x) - g(x))^2. The volume is the definite integral of this area formula.

The volume of a solid is calculated by the definite integral V = ∫[c,d] A(y) dy. What does this form of the integral imply about the orientation of the cross-sections?

A) The cross-sections are parallel to the xy-plane.

B) The cross-sections are perpendicular to the x-axis.

C) The cross-sections are perpendicular to the y-axis.

D) The cross-sections are circular, forming a volume of revolution.

Correct Answer: C

Integrating with respect to y (using dy) means that we are summing up the areas of thin slices along the y-axis. For this to work, the cross-sections must be perpendicular to the y-axis, with their area A(y) expressed as a function of y.

A student is setting up an integral to find the volume of a solid with square cross-sections. The side length of the square at any point x is (3-x). The student writes the integrand as (3-x). Why is this incorrect?

A) The student should have used the formula for the perimeter of the square.

B) The student used the side length as the integrand instead of the area.

C) The integrand should be π(3-x)^2.

D) The student should have integrated with respect to y.

Correct Answer: B

The content specifies using the area formulas for the cross-sections. The volume is the integral of the cross-sectional area. For a square with side length s = (3-x), the area is A = s^2 = (3-x)^2. The student's integrand, (3-x), represents the side length, not the area, and would calculate the area under the line y=3-x, not the volume of the solid.

A solid has a base in the xy-plane. Cross-sections perpendicular to the x-axis are rectangles with a constant height of 5. The base of each rectangle lies in the xy-plane and has a length given by s(x) over the interval [a, b]. Which integral represents the volume of this solid?

A) ∫[a,b] (s(x) + 5) dx

B) ∫[a,b] 5[s(x)]^2 dx

C) ∫[a,b] 5s(x) dx

D) ∫[a,b] 2(s(x) + 5) dx

Correct Answer: C

The volume is the integral of the cross-sectional area, A(x). The cross-sections are rectangles with area = base × height. The base is given by s(x) and the height is a constant 5. Therefore, the area of a cross-section is A(x) = 5 * s(x). The volume is the definite integral of this area function.

To find the volume of a solid with square cross-sections perpendicular to the y-axis, the side length of a square is determined by the horizontal distance between two curves, x = f(y) and x = g(y), where f(y) ≥ g(y). What is the correct expression for the area, A(y), of a single cross-section?

A) f(y) - g(y)

B) (f(y))^2 - (g(y))^2

C) π(f(y) - g(y))^2

D) (f(y) - g(y))^2

Correct Answer: D

The side length, s(y), of the square is the horizontal distance between the curves, which is f(y) - g(y). The content requires using the area formula for the shape. The area of a square is side squared. Therefore, the area of a single cross-section is A(y) = [s(y)]^2 = (f(y) - g(y))^2.