Parametric Equations

This one will be a short post, it's mostly intuition as opposed to a definite proof.

So I was musing over finding the area under a parametric curve. And I considered how we'd usually find the area under a curve, integration. Which in 2D with an x and y axis can be expressed generally as:
\[ \int_a^b y \ \mathrm{d}x\]
Parametric equations define a curve in terms of a parameter, usually t. So let's consider an arbitrary set of parametric equations:
\[\begin{align*} x=x(t) \\ y = y(t) \end{align*} \]
So to change the dx to a dt to allow integration one must differentiate x with respect to t and solve for dx.
\[ \begin{align*} x &= x(t) \\ \frac{\mathrm{d}x}{\mathrm{d}t} &=x'(t)\\ \mathrm{d}x &= x'(t) \mathrm{d}t \end{align*}\]
The value for dx must then be substituted back into the original expression.
\[\int_a^b y(t)x'(t) \ \mathrm{d}t\]
And that's it, how to find the area under a curve defined by a parametric equation.