Given the curves $y=x^2$ and $y=4$, find the area of the region bounded by these curves between $x=-2$ and $x=2$.

Find the shaded area between $f\left(x\right)=x^3+2x^2$ & $g\left(x\right)=x+2$.

Find the area of the shaded region between $f\left(x\right)=\sin 2x$ & $g\left(x\right)=2\sin x$ from $x=0$ to $x=2\pi$.

Let R be the region bounded by the graphs of $y=2x$ and $y=4x-x^2$. What is the area of R?

Given the region bounded above by $y=x^2$ and below by $y=x$ on the interval $0\le x\le 1$, determine the $x$- and $y$-coordinates of the centroid of the shaded area.

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.