To set up:
An integral for the volume of a solid.
If x is the radius of a typical shell, then circumference=2πx and height is y=f(x).
Then by shell method, the volume of the solid obtained by rotating the region under the curve y=f(x) about y- axis from a to b is
where 2πx is circumference, fx is height, and dx is the thickness of the shell.
y=cos2x, x≤π2, y= 14;about x=π2
The region is bounded by
y=cos2x, x≤π2, y= 14; about x=π2
Draw the graph of the given curves.
The graph shows the region and representative strip to rotate about the line
It has radius
Using shell method, find the approximating shell with radius
Therefore, circumference is
and height is
y=cos2x & y=14
cos2x=14 ⇔ cosx=±12 ⇔x=±π3±2nπ
a=-π3 and b=π3
So the volume of the given solid is
The integral for the volume of a solid obtained by rotating the region bounded by the given curves about the line