Integrating for a Curve in Parametric Form

The area under a curvebetween the limitsandis given by the integralIf however,andare given as functions of a parametera lot more work may be required. Instead of integrating with respect towe may integrate with respect to t usinghence

Example: A curve is given by the parametric equationsFind the area under the curve between the values

Example: A curve is given by the parametric equationsFind the area under the curve between the values

We expand the brackets to obtainThe integral becomes

To evaluate this integral we rearrange the identityto give

This method has the advantage of making the integral of many closed curves much simpler, since they may often be parametrized in terms of a parameter that varies between 0 and