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).
By shell method, the volume of the solid obtained by rotating the region under the curve y=f(x) from a to b about y- axis is
where 2πx is circumference, fx is height, and dx is the thickness of the shell.
about the y axis.
The equation of given curves is
The region and typical shell are shown in the figure.
The graph shows the region and height of cylindrical shell formed by rotation about the y-axis.
It has radius =x
Using shell method, find the typical approximating shell with radius x.
Therefore, circumference is 2πx and height is
The curves y=tanx,y=x intersect at 0,0.
The curves y=x
intersect at π3,π3
a=0 and b=π3
So the volume of given solid is
V= ∫0π32πxtan(x)-x dx
The integral for the volume of a solid obtained by rotating the region bounded by the given curves about the y-axis is