Question

Find the volume of the solid obtained by rotating the region bounded by y = x 3 , y = 0 and x = 1 about the line y = −3.

Answer 1

A cross-section is a washer with inner radius
`3` and outer radius
`x^3 + 3`


`V = \int_0^1 A(x)\, dx = \int_0^1 \pi\left[ (x^3+3)^2 - 3^2 \right]dx = \int_0^1 \pi\left( x^6 + 6x^3 \right)dx \\ \\ = \pi\left[ (1)/(7)x^7 + (3)/(2)x^4 \right]_0^1 = \pi\left((1)/(7)+ (3)/(2) \right) = (23)/(14)\pi`