Area under a parametric curve

 Goal: Finding the area under the curve of a parametric equation

Area under parametric curve

Let's consider the the area of the curve bounded by the curve y = f(x), the x-axis and the verticals x = a and x = b

We know that the area is given by:

We assume that the curve is given by the parametric equations: x = x(t) y = y(t). The formula above becomes:

The formula that allows to find the area under a parametric curve is then given by:

Example

Find the area under curve of the cycloid define by the equations:

x(t) = t-sint   y(t) = 1-cost  0  ≤ t ≤ 2π

Solution

Using the formula we have:

Practice

Find the area under the curve of the hypocycloid defined by the following equations:

x(t) = 3cost + cos3t   y(t) = 3sint - sin3t  0 ≤  t ≤ π