37–43 The region bounded by the given

Question

37–43 The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. 43.x=(y-1)^2,x-y=1 about x=-1

Answer 1

First figure out bounds of the bounded region. The curves intersect when (y-1)² = 1+y which happens at points (1,0) and (4,3).

Looking at the curves, it looks like a washer approach is best. The washers will be integrated in y with rotation radius being x(y)-(-1). The outer radius will be x(y) = 1+y and the inner radius will be x(y) = (y-1)²

The volume of rotation integral: Integral between y limits of (πr₀²- πr_i²) dy or

Integral from 0 to 4 of π((y+1)+1)² - ((y-1)²+1)²)dy = integral from 0 to 4 of π((y+2)² + (y²-2y+2)²)dy

I'll leave the integral to you. Take care.