In this section we will find a formula for determining the
area under a parametric curve given by the parametric equations,
We will also need to further add in the assumption that the
curve is traced out exactly once as t
increases from α to β.
We will do this in much the same way that we found the first
derivative in the previous section. We
will first recall how to find the area under on .
We will now think of the parametric equation as a substitution in the integral. We will also assume that and for the purposes of this formula. There is actually no reason to assume that
this will always be the case and so we’ll give a corresponding formula later if
it’s the opposite case ( and ).
So, if this is going to be a substitution we’ll need,
Plugging this into the area formula above and making sure to
change the limits to their corresponding t
values gives us,
Since we don’t know what F(x)
is we’ll use the fact that
and we arrive at the formula that we want.
Area Under Parametric
Curve, Formula I
Now, if we should happen to have and the formula would be,
Area Under Parametric
Curve, Formula II
Let’s work an example.
Example 1 Determine
the area under the parametric curve given by the following parametric equations.
First, notice that we’ve switched the parameter to θ for this problem. This is to make sure that we don’t get too
locked into always having t as the
Now, we could graph this to verify that the curve is
traced out exactly once for the given range if we wanted to. We are going to be looking at this curve in
more detail after this example so we won’t sketch its graph here.
There really isn’t too much to this example other than
plugging the parametric equations into the formula. We’ll first need the derivative of the
parametric equation for x however.
The area is then,
The parametric curve (without the limits) we used in the
previous example is called a cycloid. In its general form the cycloid is,
The cycloid represents the following situation. Consider a
wheel of radius r. Let the point where the wheel touches the
ground initially be called P. Then start rolling the wheel to the
right. As the wheel rolls to the right
trace out the path of the point P. The path that the point P traces out is called a cycloid and is given by the equations
above. In these equations we can think
of θ as the angle through which the point P has rotated.
Here is a cycloid sketched out with the wheel shown at
various places. The blue dot is the
point P on the wheel that we’re using
to trace out the curve.
From this sketch we can see that one arch of the cycloid is
traced out in the range . This makes sense when you consider that the
point P will be back on the ground
after it has rotated through an angle of 2π.