The volume of a solid formed by rotating the curve x = g(y) about the y-axis is given by which integral?

The volume of a solid formed by rotating the curve x = g(y) about the y-axis is given by which integral?

• A. π ∫ y^2 dx
• B. π ∫ g(y)^2 dy
• C. π ∫ x^2 dy
• D. 2π ∫ x dy

Answer: (C)

Rotating a region about a vertical axis and using discs means each cross-section perpendicular to the axis is a circle whose radius is the distance to the axis. Here that distance is x, and since x = g(y), the radius of a disk at height y is g(y). A thin slice at height y contributes a volume dV = area × thickness = π[g(y)]^2 dy. Integrating over the y-range of the region gives V = π ∫ g(y)^2 dy. Because x is the radius, you can also write this as π ∫ x^2 dy, with x understood as the function g(y). That matches the intended form. The other forms don’t align with using discs around the y-axis or with the radius being the x-value.

Rotating a region about a vertical axis and using discs means each cross-section perpendicular to the axis is a circle whose radius is the distance to the axis. Here that distance is x, and since x = g(y), the radius of a disk at height y is g(y).

A thin slice at height y contributes a volume dV = area × thickness = π[g(y)]^2 dy. Integrating over the y-range of the region gives V = π ∫ g(y)^2 dy. Because x is the radius, you can also write this as π ∫ x^2 dy, with x understood as the function g(y). That matches the intended form. The other forms don’t align with using discs around the y-axis or with the radius being the x-value.