Volume Using Washer Method

Calculus 1 · Axiom Academy

EXAMPLE Volume Using Washer Method When the rotated region has a gap from the axis, each slice is a washer: outer radius squared minus inner radius squared. The region bounded by y = x and y = x^2 is rotated about the x -axis. Because the region sits above the axis with a gap underneath, every cross-section is a washer (a disk with a hole). Find the volume of the resulting solid. Nice work — you built the whole washer integral and evaluated it to . Washer = disk with a hole: each cross-section has area , where R is the outer radius and r is the inner radius. Square first, then subtract: you subtract r^2 from R^2 — not (R-r)^2 . Here that was x^2 - (x^2)^2 = x^2 - x^4 . When to reach for it: use washers whenever the rotated region does not touch the axis, so a gap (the hole) runs through the solid. The washer method is just the disk method with the inner piece carved out — same integral, one extra radius to track.

This is the written version of the interactive lesson above. See the full Calculus 1 course.