#### To determine

**To find:**

The volume of the circular cone with the height h and the base radius r.

#### Answer

πr2h3unit3

#### Explanation

**1) Concept:**

i. If x is the radius of the typical shell, then the circumference =2πx and the height is y.

ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about y- axis from a to b is

V= ∫ab2πx f(x)dx

where, 0≤a≤b

**2) Given:**

The right circular cone with the height h and the base radius r

**3) Calculation:**

The right circular cone with the height h and the base radius r is as shown in the figure. It is obtained by rotating a triangle joining (0,0), (0, h) and (r,0) about y-axis

From the figure, write the equation of line passing through the two points (r, 0) and (0, h).

The slope of the line is =h-00-r The y-intercept is h

Therefore, the equation of the line is y=-hrx+h, and it rotates about y-axis.

Using the shell method, find the typical approximating shell with the radius x.

Therefore, the circumference is 2πx and the height is y=-hrx+h

So, the total volume is

V= ∫ab2πx-hrx+hdx

V= 2πh∫0r-x2r+xdx

V=2πh-x33r+x220r

V=2πh-r33r+r22-0

V=2πh -r23+r22-0

=2πhr26

V=πr2h3

Therefore, the volume of the circular cone of the height h and the radius r is V=πr2h3unit3

**Conclusion:**

The volume of the cone of the height h and the radius r is

V=πr2h3unit3