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AP Calculus AB Practice Quiz: Volume with Washer Method: Revolving Around Other Axes

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

Let R be the region in the first quadrant enclosed by the graphs of y = √x and y = x². Which of the following definite integrals gives the volume of the solid generated when R is revolved about the horizontal line y = -1?

All Questions (7)

Let R be the region in the first quadrant enclosed by the graphs of y = √x and y = x². Which of the following definite integrals gives the volume of the solid generated when R is revolved about the horizontal line y = -1?

A) π ∫[0, 1] ((√x + 1)² - (x² + 1)²) dx

B) π ∫[0, 1] ((√x - 1)² - (x² - 1)²) dx

C) π ∫[0, 1] (√x - x²)² dx

D) π ∫[0, 1] ((1 - x²)² - (1 - √x)²) dx

Correct Answer: A

The region is bounded by the curves from their intersection points at x=0 and x=1. When revolving around a horizontal line y = k, the volume is given by V = π ∫[a, b] (R(x)² - r(x)²) dx. The axis of revolution is y = -1. The outer radius R(x) is the distance from the axis to the farther curve (y = √x), so R(x) = √x - (-1) = √x + 1. The inner radius r(x) is the distance from the axis to the closer curve (y = x²), so r(x) = x² - (-1) = x² + 1. Therefore, the integral is π ∫[0, 1] ((√x + 1)² - (x² + 1)²) dx.

Let R be the region bounded by the graphs of y = 4 - x² and the x-axis. Which of the following definite integrals represents the volume of the solid generated by revolving R about the horizontal line y = 5?

A) π ∫[-2, 2] (5 - (4 - x²))² dx

B) π ∫[-2, 2] ((4 - x²) - 5)² dx

C) π ∫[-2, 2] (5² - (1 + x²)²) dx

D) π ∫[-2, 2] (5² - (4 - x²)²) dx

Correct Answer: C

The region is bounded by y = 4 - x² and y = 0, which intersect at x = -2 and x = 2. The axis of revolution is y = 5, which is above the region. The outer radius R(x) is the distance from the axis y=5 to the farther boundary, y=0. So, R(x) = 5 - 0 = 5. The inner radius r(x) is the distance from the axis y=5 to the closer boundary, y = 4 - x². So, r(x) = 5 - (4 - x²) = 1 + x². The volume is given by the washer method formula V = π ∫[a, b] (R(x)² - r(x)²) dx, which becomes π ∫[-2, 2] (5² - (1 + x²)²) dx.

Let R be the region bounded by the graphs of x = y² and x = 4. What is the volume of the solid generated when R is revolved about the vertical line x = 5?

A) π ∫[-2, 2] (5² - (y²)²) dy

B) π ∫[-2, 2] ((5 - y²)² - 1²) dy

C) π ∫[0, 4] ((5 - √x)² - 1²) dx

D) π ∫[-2, 2] (4² - (y²)²) dy

Correct Answer: B

Since the revolution is around a vertical line, we integrate with respect to y. The region is bounded by x = y² and x = 4, which intersect at y = -2 and y = 2. The axis of revolution is x = 5. The outer radius R(y) is the distance from the axis x=5 to the farther curve, x = y². So, R(y) = 5 - y². The inner radius r(y) is the distance from the axis x=5 to the closer curve, x = 4. So, r(y) = 5 - 4 = 1. The volume is V = π ∫[c, d] (R(y)² - r(y)²) dy, which is π ∫[-2, 2] ((5 - y²)² - 1²) dy.

The region R is bounded by the graph of y = sin(x) and the x-axis for 0 ≤ x ≤ π. Which of the following integrals gives the volume of the solid generated by revolving R about the line y = -1?

A) π ∫[0, π] (sin(x) + 1)² dx

B) π ∫[0, π] (sin²(x) - 1) dx

C) π ∫[0, π] ((sin(x) + 1)² - 1²) dx

D) π ∫[0, π] (sin(x) - 1)² dx

Correct Answer: C

The solid has washer-shaped cross-sections. The axis of revolution is y = -1. The outer radius R(x) is the distance from the axis to the outer curve y = sin(x), which is R(x) = sin(x) - (-1) = sin(x) + 1. The inner radius r(x) is the distance from the axis to the inner curve y = 0 (the x-axis), which is r(x) = 0 - (-1) = 1. The volume is given by V = π ∫[a, b] (R(x)² - r(x)²) dx. For the given region, this is π ∫[0, π] ((sin(x) + 1)² - 1²) dx.

Let R be the region in the first quadrant bounded by y = x, y = 2, and the y-axis. Which definite integral gives the volume of the solid formed by revolving R about the vertical line x = 3?

A) π ∫[0, 2] (3² - (3 - y)²) dy

B) π ∫[0, 2] ((3 - y)² - 3²) dy

C) π ∫[0, 2] (2² - y²) dy

D) π ∫[0, 2] (3 - y)² dy

Correct Answer: A

For revolution around a vertical line, we integrate with respect to y. The region is bounded on the right by x = y (from y=x) and on the left by x = 0 (the y-axis), from y = 0 to y = 2. The axis of revolution is x = 3. The outer radius R(y) is the distance from x=3 to the farther boundary, x = 0. So, R(y) = 3 - 0 = 3. The inner radius r(y) is the distance from x=3 to the closer boundary, x = y. So, r(y) = 3 - y. The volume is V = π ∫[c, d] (R(y)² - r(y)²) dy, which is π ∫[0, 2] (3² - (3 - y)²) dy.

Let R be the region enclosed by the graphs of f(x) and g(x), where f(x) ≥ g(x) for a ≤ x ≤ b. The volume of the solid formed by revolving R about the horizontal line y = k, where k > f(x) for all x in [a, b], is given by which integral?

A) π ∫[a, b] ((f(x) - k)² - (g(x) - k)²) dx

B) π ∫[a, b] ((k - f(x))² - (k - g(x))²) dx

C) π ∫[a, b] ((k - g(x))² - (k - f(x))²) dx

D) π ∫[a, b] (f(x) - g(x))² dx

Correct Answer: C

The axis of revolution y = k is above the entire region. The volume is found using the washer method: V = π ∫[a, b] (R(x)² - r(x)²) dx. The outer radius R(x) is the distance from the axis y=k to the farther boundary, which is y = g(x). Thus, R(x) = k - g(x). The inner radius r(x) is the distance from the axis y=k to the closer boundary, which is y = f(x). Thus, r(x) = k - f(x). Substituting these into the formula gives V = π ∫[a, b] ((k - g(x))² - (k - f(x))²) dx.

To find the volume of a solid generated by revolving a region bounded by y = f(x) and y = g(x) around a horizontal line y = c, the washer method is used. The integrand in the definite integral is π(R(x)² - r(x)²). How are the outer radius R(x) and inner radius r(x) determined?

A) R(x) = f(x) and r(x) = g(x), regardless of the axis of revolution.

B) R(x) is the distance from the axis of revolution to the curve farther from the axis, and r(x) is the distance from the axis to the curve closer to the axis.

C) R(x) is the distance from the x-axis to the top curve, and r(x) is the distance from the x-axis to the bottom curve.

D) R(x) = f(x) - c and r(x) = g(x) - c, regardless of which function is farther from the axis.

Correct Answer: B

The fundamental principle of the washer method when revolving around any axis (horizontal or vertical) is that the radii are measured from that axis of revolution. The outer radius, R, is the distance from the axis to the boundary of the region that is farther away. The inner radius, r, is the distance from the axis to the boundary of the region that is closer. The volume is then calculated by integrating the area of the resulting washer, π(R² - r²).