Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

Answer 1

Step-by-step explanation:

A cross-section is a washer with an inner radius of 8sin(x) - (-1) and an outer radius of 8cos(x) - -(1), so its area would be:

A(x) = π[(8cos(x) + 1)^2 − (8sin(x) + 1)^2]

= π[64cos^2(x) + 16cos(x) + 1 - 64sin^2(x) − 16sin(x) − 1]

= π[64cos(2x) + 16cos(x) - 16sin(x)]

=> V(x) = ∫[0,π/4] π[64cos(2x) + 16cos(x) - 16sin(x)] dx

= π[32sin(2x) + 16sin(x) + 16cos(x)] |[0,π/4]

= π[32sin(π/2) + 16√2/2 + 16√2/2 - 16]

= π(32 - 16 + 16√2) = π(16 + 16√2)

The volume of the region is π(16 + 16√2).