Shell Method (BC)

AP Calculus · Axiom Academy

An alternative approach using cylindrical shells Sometimes the disc/washer method forces you to solve for one variable in terms of another, making the integral awkward or impossible. The shell method lets you integrate with respect to whichever variable is most natural. Imagine peeling a solid of revolution into thin, nested cylindrical shells — like the layers of an onion. Each shell is a thin hollow cylinder with a certain radius, height, and thickness. When rotating the region under about the y-axis , each vertical strip at position x becomes a cylindrical shell: Radius = x (distance from the strip to the y-axis) Height = f(x) (the function value at that strip) Each shell contributes a volume of Step 3: Shells vs. Discs — When to Use Which Both methods always give the same answer, but one is usually much easier than the other. The decision depends on the axis of rotation and how the function is expressed. Rotation about y-axis with y = f(x): (discs — must solve for x = g(y), integrate w.r.t. y) Rotation about x-axis with x = g(y): Step 4: Worked Example — Shells Problem: Find the volume obtained by rotating from x = 0 to x = 2 about the y-axis . Step 1: Identify shell components at position x: Radius = x, Height = , Thickness = dx Step 3: Simplify the integrand: Step 5: Comparison — Same Problem with Discs Let's verify by solving the same problem using the disc method . Since we rotate about the y-axis, discs are horizontal slices, so we need x as a function of y.

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