Use the Shell or Disk method (whichever is easier) to compute the volume of the solid?

first

the area under the curve is

integral from x = 0 to 1 (written as int_0^1)

int_0^1 (x-x^15)dx

Now, to get the volume about the y axis we multiply the area (above) by a differential element

x dθ and integrate from 0 to 2π

int_0^2π int_0^1 (x-x^15) x dxdθ

to get the volume about the x axis we multiply the area (above) by a differential element

1/2 y dθ and integrate from 0 to 2π

int_0^2π int_0^1 (x-x^15)*1/2 (x-x^15) dxdθ

Draw each of the differential elements to see there shape. The shell element looks like rectangle as dx -> 0

The disk element looks like a piece of pie or a triangle as dx->0. This is where the 1/2 comes from.

Disk method int x=1 - 5 pi y^2dx = pi 10/x^4 dx -10/3x^3 10/3 ( 1 - 1/5^3)