A solid is formed by rotating the region bounded by y = x − x^2 and y = 0 about the line x = 2 . Use the shell method to find the volume of the solid.

Question

Use the shell method to find the volume.

Answer 1

USING THE SHELL METHOD

V -=∫₀¹ 2π (2 −x ) ( x −x² ) d x = π / 2

USING THE DISC METHOD.

We need to find the inner and outer radius of revolution in terms of y.

Let us solve the given equation x² −x + y = 0

x = 1/2 ±√(1−4y) / 2

OUTER RADIUS = 2 − [ 1/2 −√(1−4y) / 2 ] = 3/2 + √(1−4y) / 2

INNER RADIUS = 2 − [ 1/2 +√(1−4y) / 2 ] = 3/2 − √(1−4y) / 2

Then volume = π ∫₀^1/4{ [3/2 + √(1−4y) / 2]² − [ 3/2 − √(1−4y) / 2]² } d y = π/2