Volumes of Solid of  Revolution

 

If a curve is rotated about an axis which it
does not cut, the area between the curve and
the axis will generate a  solid of revolution.

The volume generated is the limit of the sums
of the volumes of cylinders formed by
rotating rectangles.

 

If rectangle dx.y is rotated, a cylinder is formed
with height dy and radius y.

 

The volume of this cylinder is 

The volume of the solid of revolution is

 

Example

Find the volume and name the shape generated
by the rotation through 360˚ of the  positive branch
of the circle with equation  x²+ y²= a²  about the x-axis.

 

 

Volume

 

Example

The portion of the curve y=2x² between y=1 and y=3
is rotated  through 360˚  about the y-axis.
Find the volume of the resulting solid.

 

Volume