calculus2

Use cylindrical shells to find the volume V of the solid.

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1. Let a circle x^2 + y^2 = r^2 be rotated about the x-axis to create the sphere
so we need
volume = π∫ y^2 dx from -r to r or
volume = 2π∫ y^2 dx from 0 to r because of the symmetry
= 2π∫ r^2 - x^2 dx from 0 to r
= 2π [r^2 x - (1/3)x^3 ] from 0 to r
= 2π (r^3 - r^3/3)
= 2π(2/3) r^3
= (4/3)π r^3

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