Parametric area

For the curve defined by $x(t)=2t, y(t)=t^2$ from $t=0$ to $3$, what is the area under the curve?

Find the area of one arch of the cycloid $x=\theta-\sin\theta$, $y=1-\cos\theta$, between $\theta=0$ and $\theta=2\pi$.

For a parametric curve $x=x(t)$, $y=y(t)$, why is the area formula $\int y\,dx$ rewritten as $\int y\,\dfrac{dx}{dt}\,dt$ rather than integrated directly in $x$?