#### To determine

**To find:**

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

#### Answer

πr2h3.

#### Explanation

**1) Concept:**

Use the definition of Volume

**2) Definition of volume:**

Let S be a solid that lies between x=a and x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is continuous function, then the volume of S is

V=limn→∞∑i=1nAxi*∆x=∫abAxdx

**3) Given:**

Cone with height h, and base radius r.

**4) Calculations:**

Consider the cone positioned about x-axis, as shown in the diagram.

At x, a slice through the cone (cross section) is circular disk. Let it’s radius be R.

By property of similar triangles,

Rx=rh →R=rhx

Hence, the area of that cross sectional slice is

A=πR2=πrhx2=πr2h2x2.

Integrating this function from 0 to h will yield the volume of the required cone

V=∫abAxdx=∫0hπr2h2x2dx=πr2h2∫0hx2dx

After integrating,

V=πr2h2x33h0

By the fundamental theorem of Calculus,

V=πr2h2h33-0

V=πr2h3

**Conclusion:**

The volume of a right circular cone with height h and base radius r is πr2h3