Area of parametric equations

Normally we know that the area under the curve from $a$ to $b$ is $\int_{a}^{b} f(x)dx$. However, what about parametric equations?
Let the curve be defined by the parametric equations $x=f(t)$, $y=g(t)$ and let the value of $t$ be increasing from $\alpha$ to $\beta$. Then we say that the area under the parametric curve is:

$A = \int_{a}^{b} y \; dx=\int_{\alpha}^{\beta} g(t)f'(t)dt$

However, if the value of $t$ is increasing from $\beta$ to $\alpha$ instead, then the area under the parametric curve will be:

$A = \int_{a}^{b} y \; dx=\int_{\beta}^{\alpha} g(t)f'(t)dt$

Be careful when determining which one to use!