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AP Calculus AB 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

The base of a solid is the region in the first quadrant bounded by the graph of y = √x, the x-axis, and the line x = 4. For this solid, each cross section perpendicular to the x-axis is a square. Which of the following definite integrals gives the volume of the solid?

All Questions (7)

The base of a solid is the region in the first quadrant bounded by the graph of y = √x, the x-axis, and the line x = 4. For this solid, each cross section perpendicular to the x-axis is a square. Which of the following definite integrals gives the volume of the solid?

A) ∫₀⁴ x dx

B) ∫₀⁴ √x dx

C) ∫₀² y⁴ dy

D) ∫₀⁴ x² dx

Correct Answer: A

The volume of a solid with known cross sections perpendicular to the x-axis is given by the integral V = ∫[a,b] A(x) dx, where A(x) is the area of a cross section. The side length, s(x), of a square cross section at a given x is the vertical distance from the x-axis (y=0) to the curve y = √x, so s(x) = √x. The area of the square is A(x) = (s(x))² = (√x)² = x. The bounds of integration are given by the line x=4 and the start of the region at x=0. Therefore, the volume is ∫₀⁴ x dx.

Let R be the region enclosed by the graphs of y = x and y = x². The base of a solid is the region R. The cross sections of the solid perpendicular to the x-axis are squares. Which integral represents the volume of the solid?

A) ∫₀¹ (x - x²) dx

B) ∫₀¹ (x² - x)² dx

C) ∫₀¹ (x - x²)² dx

D) π∫₀¹ (x - x²)² dx

Correct Answer: C

First, find the points of intersection: x = x² gives x=0 and x=1, which are the limits of integration. For a cross section perpendicular to the x-axis, the side length s(x) of the square is the vertical distance between the upper curve (y=x) and the lower curve (y=x²). So, s(x) = x - x². The area of a square cross section is A(x) = (s(x))² = (x - x²)². The volume is the integral of the area, V = ∫₀¹ (x - x²)² dx.

The base of a solid is the region enclosed by the graph of x = y², the y-axis, and the line y = 2. If the cross sections perpendicular to the y-axis are squares, what is the volume of the solid?

A) ∫₀² y² dy

B) ∫₀⁴ x dx

C) ∫₀² y⁴ dy

D) ∫₀² (√x)² dx

Correct Answer: C

Since the cross sections are perpendicular to the y-axis, we must integrate with respect to y. The limits of integration are from y=0 to y=2. The side length, s(y), of a square cross section is the horizontal distance from the y-axis (x=0) to the curve x = y². Thus, s(y) = y² - 0 = y². The area of a square cross section is A(y) = (s(y))² = (y²)² = y⁴. The volume is given by the definite integral V = ∫₀² y⁴ dy.

The base of a solid is the region in the first quadrant bounded by y = cos(x), y = sin(x), and the y-axis. Cross sections perpendicular to the x-axis are rectangles whose height is 5 times the length of their base. Which integral gives the volume of the solid?

A) ∫₀^(π/4) 5(cos(x) - sin(x))² dx

B) ∫₀^(π/4) (5(cos(x) - sin(x)))² dx

C) ∫₀^(π/4) 5(cos(x) - sin(x)) dx

D) ∫₀^(π/2) 5(sin(x) - cos(x))² dx

Correct Answer: A

The intersection of y=cos(x) and y=sin(x) in the first quadrant is at x=π/4. From x=0 to x=π/4, cos(x) ≥ sin(x). The base of a rectangular cross section is b(x) = cos(x) - sin(x). The height is given as 5 times the base, so h(x) = 5(cos(x) - sin(x)). The area of the rectangle is A(x) = b(x) * h(x) = (cos(x) - sin(x)) * 5(cos(x) - sin(x)) = 5(cos(x) - sin(x))². The volume is the integral of this area from x=0 to x=π/4.

The volume of a solid is given by the definite integral ∫[a,b] (f(x) - g(x))² dx, where f(x) ≥ g(x) on the interval [a, b]. This solid is formed by cross sections perpendicular to the x-axis on a base region bounded by y=f(x) and y=g(x). What is the shape of each cross section?

A) A rectangle with a constant height of 1

B) A square

C) A semicircle

D) An equilateral triangle

Correct Answer: B

The general formula for the volume of a solid with known cross sections is V = ∫[a,b] A(x) dx, where A(x) is the area of the cross section. In this case, the integrand is (f(x) - g(x))². The term f(x) - g(x) represents the distance between the two curves, which is the side length, s(x), of the cross section. The area is therefore A(x) = (s(x))² = (f(x) - g(x))². The area formula A = s² corresponds to a square.

The base of a solid is the region R bounded by the graphs of y = ln(x), the line x = e, and the x-axis. The solid is formed by cross sections perpendicular to the x-axis that are rectangles with a constant height of 3. Which of the following integrals represents the volume of the solid?

A) ∫₁ᵉ 3(ln(x))² dx

B) ∫₁ᵉ 9(ln(x))² dx

C) ∫₁ᵉ 3 ln(x) dx

D) ∫₀¹ 3(e - eʸ) dy

Correct Answer: C

The region is bounded by y=ln(x), y=0 (the x-axis), and x=e. The lower bound for x is where ln(x)=0, which is x=1. The cross sections are perpendicular to the x-axis, so we integrate with respect to x from 1 to e. The base of each rectangular cross section is the vertical distance b(x) = ln(x) - 0 = ln(x). The height is a constant h=3. The area of a cross section is A(x) = base × height = ln(x) * 3 = 3ln(x). The volume is the definite integral of the area: V = ∫₁ᵉ 3 ln(x) dx.

The base of a solid is the region in the first quadrant bounded by y = 4 - x² . The cross sections of the solid perpendicular to the y-axis are squares. Which of the following integrals gives the volume of the solid?

A) ∫₀² (4 - x²)² dx

B) ∫₀⁴ (4 - y) dy

C) ∫₀⁴ √(4 - y) dy

D) ∫₀² (4 - y)² dy

Correct Answer: B

Since the cross sections are perpendicular to the y-axis, we must express the boundary in terms of y and integrate with respect to y. The equation y = 4 - x² becomes x² = 4 - y, so x = √(4 - y) (since we are in the first quadrant). The side length of the square cross section, s(y), is the horizontal distance from the y-axis (x=0) to the curve, so s(y) = √(4 - y). The area of the square is A(y) = (s(y))² = (√(4 - y))² = 4 - y. The limits of integration for y are from y=0 to the y-intercept at y=4. Therefore, the volume is V = ∫₀⁴ (4 - y) dy.