between
the limits
and
is
given by the integral
If
however,
and
are
given as functions of a parameter
a
lot more work may be required. Instead of integrating with respect
to
we
may integrate with respect to t using
hence
A Level Maths Notes: C4 – Integrating when x and y are in Parametric Form
The area under a
curvebetween
the limitsandis
given by the integral
If
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 equations
Find
the area under the curve between the values
Example: A curve is given by
the parametric equations
Find
the area under the curve between the values
We expand the brackets to
obtain
The
integral becomes
To evaluate this integral we
rearrange the identity
to
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