Question

The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
x = (y − 6)^2, x = 1; about y = 5

Answer 1

Luckily, the curve intersects x=1 at y=5 and y=7, so, using shells,

v = ∫[5,7] 2πrh dy
where r=y-5 and h=1-x=1-(y-6)^2

v = 2π∫[5,7] (y-5)(1-(y-6)^2) dy
= 2π(-y^4/4 + 17y^3/3 - 9y^2/2 + 175y) [5,7]
= 8π/3