In regular integration of a function you obtain the area under by considering smaller rectangles with height and width so the approximate area is as this is written as
For a Volume of Revolution you do the same however the areas are small cylindrical volumes. These volume is in this case is the radius and we consider (the width in the case) going to zero so the approximate area as . The exact volume is written as
You can do the same thing for a volume of revolution around the y axis by rearranging to get then the volume will be
Keep in mind that and are values of and that the radius of the volume is from the axis going across parallel to the axis until the curve.
Parametric
For a parametric equation defined you can find the volumes of revolution with the following formulae:
About the x axis
About the y axis
Arc Length
Cartesian
The arc length of from and is:
Parametric
For , , the arc length from to is:
Polar
For , the arc length from half lines to is:
Surface Area of Volume of Revolution
The area, , of the surface generated when the arc on the curve is rotated completely about the -axis is , and about the -axis is .