Consider the unit circle $\left \{ \right (x,y) \in \mathbb{R}^2 : x^2+y^2\le 1\}$ 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:

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

Now since the Jacobian of the transformation is constant, the change of variables in integral theorem 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 $\pi a b$