easy calculation of the area of an ellipse

Consider the unit circle $\left\{\right(x,y)\in {ℝ}^2:x^2+y^2\le 1\}$. It’s a well known fact that the area of this set is $\pi$.

Now consider the following linear transformation $(x,y)\to (u,v)=(ax,by)$.

The determinant of the transformation is $ab$ and the transformed circle is:

$\left\{\right(u,v)\in {ℝ}^2:{\left(\frac{u}{a}\right)}^2+{\left(\frac{v}{b}\right)}^2\le 1\}$ an ellipse of axis $(a,b)$.

Now since the Jacobian of the transformation is constant, the change of variables in integral theorem (http://planetmath.org/ChangeOfVariablesInIntegralOnMathbbRn) allows us to say the area of the transformed set is $ab$times the area of the original set.

Thus, the area of an ellipse is $\pi ab$.