Area of parametric equations

In this section, we will learn find the area under the curve of parametric equations. This still involves integration, but the integrand looks changed. The integrand is now the product between the second function and the derivative of the first function. We will examine the different types of parametric equations with a given range, and learn how to find the area of each one.

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!