PrepGo

AP Calculus BC 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

According to the provided principles, which mathematical tool is fundamental for calculating the volumes of solids of revolution?

All Questions (7)

According to the provided principles, which mathematical tool is fundamental for calculating the volumes of solids of revolution?

A) Definite integrals

B) Differential equations

C) The Pythagorean theorem

D) Partial fraction decomposition

Correct Answer: A

The provided content explicitly states that one can "Calculate volumes of solids of revolution using definite integrals." This identifies definite integrals as the primary tool for this purpose.

The washer method is specifically used to find the volume of a solid of revolution when its cross-sections, taken perpendicular to the axis of revolution, have what specific shape?

A) Square

B) Ring-shaped

C) Semicircular

D) Triangular

Correct Answer: B

The content specifies that the washer method is used for volumes of solids of revolution "whose cross sections are ring shaped."

The provided information indicates that the washer method, using definite integrals, is a valid technique for finding the volume of a solid generated by revolving a region around which of the following?

A) Only the x-axis

B) Only the y-axis

C) Only lines that pass through the origin

D) Any horizontal or vertical line

Correct Answer: D

The content states that the washer method can be used to find volumes of solids of revolution "around any horizontal or vertical line," not just the primary coordinate axes.

A solid is formed by rotating a planar region about a horizontal line. If the cross-sections of this solid taken perpendicular to the axis of rotation are consistently ring-shaped, which of the following describes the appropriate method for finding the volume?

A) The washer method using definite integrals

B) A method for solids with known square cross-sections

C) The shell method using derivatives

D) An approximation using geometric formulas for cones

Correct Answer: A

The problem describes a solid of revolution around a horizontal line with ring-shaped cross-sections. The provided content directly links these characteristics to the use of "definite integrals with the washer method."

The use of the washer method implies that the cross-sections of the solid of revolution are ring-shaped. What does the geometry of a "ring" inherently suggest about the area being integrated?

A) The area is defined by a single function that touches the axis of revolution.

B) The area must be symmetrical about the axis of revolution.

C) The area is calculated as the difference between an outer boundary and an inner boundary.

D) The area can only be described using trigonometric functions.

Correct Answer: C

A ring shape is an annulus, which is the region between two concentric circles. Its area is found by subtracting the area of the inner circle from the area of the outer circle. In the context of an integral, this corresponds to integrating the area formed by the difference between an outer radius (boundary) and an inner radius (boundary).

When calculating the volume of a solid of revolution using a definite integral, the formation of ring-shaped cross-sections indicates what about the relationship between the region being revolved and the axis of revolution?

A) The region is fully bounded by the axis of revolution.

B) The axis of revolution must be one of the coordinate axes.

C) The region is tangent to the axis of revolution at exactly one point.

D) A gap exists between the region being revolved and the axis of revolution.

Correct Answer: D

A ring shape is a disk with a hole in the center. This hole is created when the region being revolved does not extend all the way to the axis of revolution. This gap between the region's inner boundary and the axis results in the ring-shaped cross-section, which necessitates the washer method.

The washer method is a specific application of using definite integrals to find volumes. The method's structure accounts for ring-shaped cross-sections. This approach is necessary because, unlike a solid disk, a ring's area is calculated by:

A) Multiplying the average radius by the circumference.

B) Finding the area of a larger circle and subtracting the area of a smaller, concentric circle.

C) Squaring the circumference and dividing by pi.

D) Integrating the function that defines the outer boundary only.

Correct Answer: B

The fundamental principle behind the washer method is based on the geometry of its ring-shaped cross-section. A ring (or washer) is the region between two circles. Its area is calculated as the area of the large outer circle minus the area of the small inner circle (the hole). The definite integral then sums the volumes of these infinitesimally thin washers.