Which of the following expressions represents the volume of a solid formed by revolving y = f(x) about the x-axis, using integration with respect to x?

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When a curve y = f(x) is revolved about the x-axis, each cross-section perpendicular to the x-axis is a disk of radius equal to the y-value at that x. The area of such a disk is π times the radius squared, so it’s π [f(x)]^2. To get the total volume, sum these disk volumes along x from a to b, giving V = π ∫_a^b [f(x)]^2 dx. Since squaring removes any sign, negative values of f(x) don’t cause issues—the radius is the positive distance from the x-axis.

So the correct setup is the integral of π times the square of the function, with respect to x. Writing the radius as y would be the same idea, π ∫ y^2 dx, as long as y denotes f(x). Expressions that integrate with respect to y or use 2π are not the disk method for revolving around the x-axis.