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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. enter image description here

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    Can you solve the problem for the special case of a unit circle? If so, the result follows by applying the change-of-variables formula to a particularly simple coordinate change. – 2017-02-20
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    No. I don't see it through, I can't solve for the unit circle. – 2017-02-20
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    If we consider the circle $x^2 + y^2 = 1 + \epsilon$ with $\epsilon$ small, we can see that the radius of the circle satisfies $r^2 = 1 + \epsilon$, so $r = 1 + \frac{1}{2} \epsilon + O(\epsilon^2)$. Thus, its area is $\pi \left(1 + \frac{1}{2} \epsilon + O(\epsilon^2)\right)^2 = \pi + \pi \epsilon + O(\epsilon^2)$, and the area of the annulus is $\pi \epsilon + O(\epsilon^2)$. – 2017-02-20
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    A uniform thickening of a noncircular ellipse is not itself an ellipse. – 2017-02-20
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    I am not thickening the ellipse. I am increasing the value of c to c+dc. That is an ellipse, isn't it? – 2017-02-20

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As the ellipse can be derived from a circle by means of a dilation along $y$ of ratio $b/a$, we can find the desired area by considering a circular ring with radii $a\sqrt c$ and $a\sqrt{c+dc}$ and then multiplying the result by $b/a$. A $y$ section of such a circular ring at $x$ has a width $$ dy=\sqrt{a^2(c+dc)-x^2}-\sqrt{a^2c-x^2}={1\over2} {a^2\over \sqrt{a^2c-x^2}}dc $$ to first order in $dc$. The area, to first order, is formed by two parallelograms of basis $dy$ and height $dx$, so we have in the end for the desired area in the ellipse: $$ dA = {b\over a}\,2\,dy\,dx={ab\over \sqrt{a^2c-x^2}}dc\,dx. $$ Notice that by integrating the above for $-a\sqrt c

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    That the area of the ring, but how do I find the area of the hatched region i.e. between $x$ and $x+dx$? – 2017-02-20
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    I hadn't read that comment in the picture: you'd better point that out in the main text of your question. – 2017-02-20
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    Just edited my answer. – 2017-02-20
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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 $$dS=ab\pi{dc}$$

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Note that you just scaled up both the major and minor axes by a common factor $ \sqrt c$

You want to look at

$$ (\sqrt c\, y )\ d (\sqrt c x ) - y\ dx = (c-1) y\ dx . $$