#### To determine

**To find:**

Volume of the remaining portion of the sphere

#### Answer

V=43πR2-r232

#### Explanation

**1) Concept:**

i) Solid of Revolution:

Value of solid revolution revolving a region around a line is given by

V=∫abA(x)dx or V=∫cdA(y)dy

In case of thedisk method, radius is found in terms of x or y and use A=π·radius2

In case of thewasher method, find theinner and outer radius and compute the area of washer by subtracting the area of inner disk from outer disk, use

A=π·outer radius2-π·inner radius2

ii) Integrals of Symmetric functions:

Suppose f is continuous on [-a, a]

a) If f is even [f-x=fx], then ∫-aaf(x)dx=2∫0af(x)dx

b) If f is odd [f-x=-fx], then ∫-aaf(x)dx=0

**2) Calculation:**

Consider that sphere of radius R is cut from thecenter along x axis as shown in the figure

The sphere looks like a circle with center as the origin. So equation of circle is

x2+y2=R2

y2=R2-x2

Since, y=r

r2=R2-x2

x2=R2-r2

x=±R2-r2

So, x varies from R2-r2 to -R2-r2

To find the volume of theremaining portion means to find the volume of the shaded region as shown below.

To find the volume, considerthe strip which is to be rotated about xis; this forms the washer with outer radius as y=R2-x2 and inner radius as r

So area can be calculated as

Ax=π·outer radius2-π·inner radius2

Ax=π·R2-x22-π·r2

The volume is

V=∫-R2-r2R2-r2π·R2-x22-π·r2dx

By using theconcept of integrals of symmetric functions

V=2∫0R2-r2π·R2-x22-π·r2dx

Factor out π,

V=2π∫0R2-r2R2-x22-r2dx

V=2π∫0R2-r2R2-x2-r2dx

V=2π∫0R2-r2R2-r2-x2dx

V=2π R2-r2 x-x330R2-r2

V=2πR2-r2R2-r2-R2-r233

V=2πR2-r2R2-r212-R2-r23/23

V=2πR2-r232-R2-r23/23

Factor out

R2-r232

V=2π·R2-r2321 -13

V=2π·R2-r23223

V=43π·R2-r232

Hence volume of remaining portion is

V=43π·R2-r232

**Conclusion:**

V=43π·R2-r232