To calculate the area beneath a parametric curve you have to use the Substitution Rule (i.e. U-Substition) to integrate $f(t)$ and $g(t)$:
If $A = \int_a^b F(x) dx$ and $y=g(t)=F(x)$, then the area under the parametric curve defined by $x=f(t), y=g(t)$ is:
$$A = \int_{\alpha}^{\beta} g(t)f'(t) dt$$where $\alpha = f(a)$ and $\beta = g(b)$.
So, in practice, you just need $y=g(t)$ and to find $\frac{dx}{dt}$ for $x=f(t)$. Then you just integrate:
$$A = \int_{\alpha}^{\beta} g(t)\frac{dx}{dt} dt$$
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