Integration is useful for finding volumes, provided the object has an axis of symmetry.
Finding Volumes using Integration
The basic idea is to cut the object up into many thin slices each perpendicular to the axis of symmetry. The area of a typical slice is then found and these are then summed by integrating between the limits of the given figure. The differential, dx, represents the thickness of the slice.
Example  
Use integration to find the volume of this squarebased pyramid. The base is 40 m by 40 m. The height is 20m. 

The pyramid is placed on a graph on its side. A typical slice is a square with side 2x. It has an area of 4x^{2} and a thickness of dx. 
A typical slice looks like this. 
Volumes of Revolution
By far the most important application of volumes and integration is finding the volume of figures with circular crosssections.
These are obtained by rotating a curve on a graph about either the xaxis or the yaxis.
Example  
The curve y = √ x with 0 < x < 4 is rotated about the xaxis. Find the volume of revolution. Here a typical slice is a circle, with radius √ x. The area of the slice is:

A typical slice looks like this. 
In general:
For volumes around the xaxis: where a and b lie on the xaxis. 
and
For volumes around the yaxis: where c and d lie on the yaxis. 
Example
The part of the line y = x − 3 between y = 0 and y = 2 is rotated about the yaxis. Find the volume of the solid generated. .y = x − 3 