#### To determine

**To find:**

The volume of a cap of a sphere with radius r and height h.

#### Answer

πh2r-h3

#### Explanation

**1) Concept:**

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 a continuous function, then the volume of S is

V=limn→∞∑i=1nAxi*∆x=∫abAxdx

**2) Given:**

Radius of sphere is r and height of cap is h.

**3) Calculation:**

Consider a cross section of the cap at height h from the origin as shown in above figure.

To find the cross section,

Consider a circle x2+y2=r2

Subtract y2 from both sides,

x2=(r2-y2)

Taking square root to find the value of x,

x=r2-y2

From the figure, limits of integral are from r-hto r.

So, the volume of the cap of sphere is

V=π∫r-hrr2-y2dy

By integrating,

V=πr2y-y33rr-h

By substituting the limits,

V=πr2·r-r33-r2r-h-r-h33

After simplification,

V=13π2r3-r-h3r2-r2-2rh+h2= πh2r-13.

**Conclusion:**

The volume of a cap of a sphere with radius r and height h is πh2r-h3