Additional Further Calculus

Reduction formulae can be used to evaluate integrals using recursive techniques. Often, they are useful for integrals with a function raised to a power, where integration by parts can then be applied.

The arc length of a curve is shown below.

The arc length of between and is given by:

Or, alternatively, for a parametric curve where and are the derivatives of and with respect to (between and ):

The formulae for arc length are given in the formulae booklet.

The surface area of revolution for resolved about the -axis between and is given by:

Or, alternatively, for a parametric curve in terms of resolved about the -axis between and :

The surface area of revolution for resolved about the -axis between and is given by:

Or, alternatively, for a parametric curve in terms of resolved about the -axis between and :

The formulae for surface areas of revolution are given in the formulae booklet.