Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., $x=x\left(t\right),y=\left(t\right),$ then the area of the figure is evaluated by one of the three formulas :
$S=-\stackrel{\beta }{\underset{\alpha }{\int }}y\left(t\right)x^{\prime }\left(t\right)dt,$
$S=\stackrel{\beta }{\underset{\alpha }{\int }}x\left(t\right)y^{\prime }\left(t\right)dt,$
$S=\frac{1}{2}\stackrel{\beta }{\underset{\alpha }{\int }}\left(x{y}^{\prime }-y{x}^{\prime }\right)dt,$
Where $\alpha \text{and}\beta$ are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t.
The area of the region bounded by an are of the cycloid $x=a\left(t-\sin t\right),y=a\left(1-\cos t\right)$ and the x-axis is

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