Find the area of an infinitesimal elliptical ring. I have an ellipse given by,
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=c$$
and another ellipse that is infinitesimally bigger than the previous ellipse, i.e.
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=c+dc$$
I want to find the area enclosed by the ring from $x$ to $x+dx$ but I don't know how. Please don't solve the question, just point me in the right direction, I want to solve it myself. Here is a picture of what I want to do.


Let $x=a\sqrt{c}\cos(\varphi)$, $y=b\sqrt{c}\sin(\varphi)$. The Jacobian is
$$J=det\begin{bmatrix}
\frac{1}{2\sqrt{c}}a\cos(\varphi) & -b\sqrt{c}\sin(\varphi) \\
\frac{1}{2\sqrt{c}}a\sin(\varphi) & b\sqrt{c}\cos(\varphi)
\end{bmatrix}=\frac{ab}{2}
$$
Hence $dS=\frac{ab}{2}dcd\varphi$, integrating over angle we get 
$$dS=ab\pi{dc}$$
