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MathCalculus

Shell Method Formula: Volume of y = 3x − x² About the y-Axis

Using shells of radius x and height 3x − x² gives V = 2π∫₀²x(3x − x²) dx = 8π cubic units.

Question

Define RR as the region bounded above by f(x)=3x−x2f(x)=3x-x^2 and below by the xx-axis over the interval [0,2][0,2].

Find the volume of the solid of revolution formed by revolving RR around the yy-axis.

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Evidence boundary

The computation uses the entire region under y = 3x − x² on [0, 2] and rotation about the y-axis. The result is exact in cubic units because the prompt gives no physical unit.