Washer Method

Calculus 1 · Axiom Academy

When the region you spin doesn't touch the axis, each slice is a disk with a hole — a washer. 1. A Reminder: Slices That Touch the Axis Are Disks Start with a region bounded by y = f(x) , the x -axis, and the lines x = a and x = b . Spin it around the x -axis. Because the region sits on the axis, every cross-section is a full circular disk , and its radius is just the height of the curve, R(x) = f(x) . Watch the slicing line sweep across — the disk it generates grows and shrinks with the curve. Each disk: radius R(x)=f(x) , thickness dx , area 2. Lift the Region Off the Axis — and a Hole Appears Now spin the region trapped between two curves, y = f(x) on top and y = g(x) below, where . The lower curve never reaches the axis, so when the solid forms there's an empty tube running through the middle. Each cross-section is a washer : an outer disk of radius R(x)=f(x) with an inner disk of radius r(x)=g(x) removed. 3. The Formula: Outer Disk Minus Inner Disk The area of one washer is the area of the outer disk minus the area of the inner disk. Slice the solid into many thin washers, sum their volumes, and let the thickness shrink — that sum becomes an integral from x=a to x=b . Watch the slices stack and the running total of climb. keeps the hole's area in the bookkeeping. treats the ring's width as a radius. Different quantity, wrong answer. 4. Putting It Together: y=x and y=x^2

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