## Integral by substitution

The trick is often using this rule backwards. For example, calculate the area of cycloid:

We know $x(t) = t - \sin t$ and $y(t) = 1 - \cos t$, We need

\begin{align*} \int_{x(2\pi)}^{x(0)} f(x) dx &= \int_{2\pi}^{0} f(x(t)) \frac{dx}{dt}dx \\ &= \int_{2\pi}^{0} y(t) \frac{dx}{dt}dx \\ &= \int_{2\pi}^{0} (1-\cos t)(1-\cos t)dx \\ \end{align*}