#### To determine

**To set up:**

An integral for the volume of a solid.

#### Answer

V=∫0π32πxtanx-xdx

#### Explanation

**1) Concept:**

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

V= ∫ab2πxf(x)dx;a≤x≤b

where 2πx is circumference, fx is height, and dx is the thickness of the shell.

**2) Given:**

y=tanx,y=x,x= π3;

about the y axis.

**3) Calculation:**

The equation of given curves is

y=tanx, y=x

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

fx=tanx-x

The curves y=tanx,y=x intersect at 0,0.

The curves y=x

and x=π3

intersect at π3,π3

So,

a=0 and b=π3

So the volume of given solid is

V= ∫0π32πxtan(x)-x dx

**Conclusion:**

The integral for the volume of a solid obtained by rotating the region bounded by the given curves about the y-axis is

V=∫0π32πx(tan(x)-x)dx