Which of the following integrals correctly represents the area of the region enclosed by the curves $y=x^2$ and $y=2x$ for $x\in [0,2]$?

Given the parametric equations $x=t^2-2t$ and $y=t$, for $t\in [0,2]$, find the area enclosed by the curve and the y-axis.

Given the parametric equations $x=t^2-2t$ and $y=t$ for $t\in [0,2]$, what is the area enclosed by the curve and the y-axis?

Find the area of the region that lies inside both curves: $r=\sin \left(2\theta \right)$ and $r=\cos \left(2\theta \right)$.

Given the parametric equations $x=t^2-3t$ and $y=t$, for $t\in [0,3]$, find the area enclosed by the curve and the y-axis.

Find the area of the region that lies inside both curves given by $r=3\cos \left(\theta \right)$ and $r=\sin \left(\theta \right)$. Which of the following is the correct area?

Find the area of the region bounded by the curves $y=\sin 2x$ and $y=\sin 3x$ for $0\le x\le \pi$.

Given the region bounded above by $y=x^2$ and below by $y=0$ for $0\le x\le 2$, what are the coordinates of the centroid of the shaded area?

Find the area of the region enclosed by the inner loop of the curve $r=2+4-\sin \theta$.