Region Between y = x and y = x² Rotated Around x-axis

Calculus 2 · Axiom Academy

EXAMPLE Region Between y = x and y = x^2 Rotated Around the x-axis Use the washer method to find the volume of the solid formed when the region is revolved about the x-axis. The region bounded by the curves y = x and y = x^2 is rotated about the x-axis. Find the volume of the resulting solid. (The curves meet at x = 0 and x = 1 , so that is the region of interest.) The shaded region lies between y = x (outer, farther from the axis) and y = x^2 (inner) from x = 0 to x = 1 . Nice work — you found the volume with the washer method. Here is what to carry forward: When to use washers: rotating a region with a gap to the axis leaves a hole, so each cross-section is a washer (an annulus), not a solid disk. Identify the radii: the outer function ( y = x ) is farther from the axis and the inner function ( y = x^2 ) is closer; over , . Bounds from intersections: setting x = x^2 gives x = 0 and x = 1 , the limits of integration. Simplify before integrating: R^2 - r^2 = x^2 - x^4 , which integrates cleanly with the power rule. The washer method is the go-to for volumes of hollow solids of revolution — always sketch the region first and confirm which curve is the outer radius.

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