Shell Method - Volume of Revolution

Calculus 1 · Axiom Academy

EXAMPLE Shell Method — Volume of Revolution Find a volume of revolution with cylindrical shells when the axis runs parallel to the strips The region under from x=0 to x=4 is rotated about the y -axis . Find the volume of the resulting solid using the shell method. Left: the region under on [0,4] , with one representative vertical strip at distance x from the y -axis. Right: spun about the y -axis, that strip sweeps out a thin cylindrical shell of radius x and height . Nice work — you built a volume of revolution one cylindrical shell at a time. The pieces worth keeping: Shell formula: — each shell is a thin can of circumference . Match the method to the axis: rotating about the y -axis while integrating in x is exactly when shells beat disks — no need to solve the curve for x . Read radius and height off the picture: here radius =x (distance to the axis) and height (the curve), so the integrand is . Cross-check: the washer method in y gives too — same solid, same answer. Disks, washers, and shells all add up infinitely many thin pieces — shells just slice the solid into nested cans instead of stacked plates. Pick whichever keeps the integral in the easier variable.

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