**Summary**

- Rotating about the
*x axis*, Volume of Revolution:

- Rotating about the
*y axis*, Volume of Revolution:

#### Definite Integrals Formula

We know that definite integrals can help us figure out areas underneath the curves.

Recalling the formula for definite integrals from the article ”Integration” :

Similarly, pretty much using the same principle we can work out volumes of rotating solids.

If the part of a curve* y = f(x)* between the giving limits *x = a* & *x = b* is rotated about the x axis or y axis through 360°, then the solid formed is called a *‘Solid of Revolution’*.

Such a solid is always symmetrical about the axis of rotation. Examples of such solids are, cone, cylinder and sphere etc.

Consider an element with radius *y* and height .

Volume of the element

Volume of the entire solid

Note: The smaller the , the approximation is closer to the actual value.

When :

Hence, we can deduce that:

The volume of revolution of* y = f(x)* between *x = a* & *x = b* about the axis of x is:

Let’s look at some examples now.

#### Example 1

Rotating about the x axis

Q. The region between the curve , the *x axis* and the line *x = 1* and *x = 3* is rotated through 360° about the *x axis*.

Find the volume of revolution which is formed.

*Solution:*

Using the formula

Replacing *y* with ,

** Ans: ** cubic units

#### Example 2

Rotating about the y axis

Q. The region between the curve , the *y axis* and the line *y = 2* and *y = 5* is rotated 360° around the* y axis*.

Find the volume of revolution obtained.

*Solution:*

Using the formula

where is now replaced by

Hence:

as

* Ans: * cubic units

##### Reference

- http://amsi.org.au/teacher_modules/Cones_Pyramids_and_Spheres.html
- MEI A Level Further Mathematics Year 2 4th Edition