Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = y^8 , y = 1 , and the y-axis about x = 5

Answer 1

x = y^8, x = 0, and y = 1 defines the region rotated about x = 5.

Then, we subtract the volume created by the x = y^8 from the volume created by x = 0.

x = y^8 meets x = 0 at y = 0.

The volume from x = 0 is a cylinder, πr²h = π5²(1-0) = 25π

The distance of x = y^8 from x = 5 is 5 - y^8

Then, using I[a,b] for the integral from a to b and E[a,b] for the evaluation from a to b, we use the disk method, and

I[0, 1] π(5 - y^8)² dy = π I[0, 1] 25 - 10y^8 + y^16 dy = π(25y - 10/9 y^9 + y^17/17) E[0,1] =

π(25 - 10/9 + 1/17 -(0)) = π(25 - 10/9 + 1/17)

Then, 25π - π(25 - 10/9 + 1/17) = 10/9π - 1/17π = 1 8/153 π or 161/153 π