# Thread: Area of an ellipse

1. ## Area of an ellipse

Just a quick query about calculating the area of an ellipse:

We've given the following parametric coordinates of an ellipse $C$:

$x = a \cos{t}$ and $y = b \sin{t}$ and asked to find the area enclosed.

It's easy enough to find the following derivatives:

$\frac{dx}{dt} = -a \sin{t}$ and $\frac{dy}{dt} = b \cos{t}$

But then in the answers it's stated that:

The area of an ellipse is given by $\frac{1}{2} \int_C x dy - y dx$

Just wondering have they worked this out, or is this a common formula?

Thanks for the help

2. Originally Posted by craig
Just a quick query about calculating the area of an ellipse:
Just wondering have they worked this out, or is this a common formula?
It is a common formula deduced from the Green's Theorem:

$\dfrac{1}{2}\displaystyle\int_{C}xdy -ydx=\dfrac{1}{2}\displaystyle\iint_{D}\left(\frac{ {\partial Q}}{{\partial x}}-\frac{{\partial P}}{{\partial y}}\right)dxdy=\dfrac{1}{2}\displaystyle\iint_{D}( 1+1)dxdy=\displaystyle\iint_{D}dxdy=A$

and $A$ is the area of $D$ .

Regards.

Fernando Revilla