#### To determine

**To find:**

The volume of the sphere of the radius r.

#### Answer

4πr33unit3

#### 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 radius of the sphere is r.

**3) Calculation:**

A sphere is obtained by rotating the region under the curve x2+y2=r2 of the first and the fourth quadrant about y-axis.

Hus we have y2=r2-x2

⇒y= ±r2-x2

Let the given region rotate about the y- axis.

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

The figure gives top half of height of a typical shell. By symmetry therefore, the circumference is 2πx and the height is y= r2-x2--r2-x2=2r2-x2

So, the total volume is

V= ∫ab2πx 2r2-x2dx

V=- 2π∫0r(-2x)r2-x212dx

Let -x2=t⇒-2xdx=dt

x=0⇒t=0 and x=r⇒t=-r2

V=- 2π∫0-r2r2+t12dt

V=-2π2r2+t3230-r2

=-2π 2r2-r2323-2r2+02323

=-2π-2r33

V=4πr33

Therefore, the volume of the sphere of the radius r is V=4πr33unit3

**Conclusion:**

The volume of the sphere of the radius r is V=4πr33unit3