Volumes by Washer Method

AP Calculus · Axiom Academy

LESSON Volumes by Washer Method Extending the disc method to solids with holes Step 1: When the Disc Method Isn't Enough The disc method works beautifully when a region bounded by a single curve is rotated about an axis. But what happens when we rotate the region between two curves ? Imagine the region between and from x = 0 to x = 1. If we rotate this region about the x-axis, the resulting solid has a hole running through its center — like a donut or pipe. A single disc can't capture this shape. Each cross-sectional slice is an annulus (ring shape) with: Outer radius R(x) = distance from axis to the farther curve Inner radius r(x) = distance from axis to the closer curve To find the total volume, we integrate the washer area along the axis of rotation, just as we did with discs. where R(x) is the outer radius (farther from axis) and r(x) is the inner radius (closer to axis). Here R(y) and r(y) are the outer and inner radii expressed as functions of y. Step 3: Setting Up a Washer Integral Before you can write down the integral, you need to identify three things: Which curve is farther from the axis? That function gives the outer radius R. Which curve is closer to the axis? That function gives the inner radius r. What are the bounds? Find where the curves intersect to determine a and b. Rotating about x-axis: Radii are vertical distances — use y = f(x) and y = g(x) directly. Rotating about y-axis: Radii are horizontal distances — solve for x in terms of y.

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