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 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 $r=2+4-\sin \theta$.

What is the area of the region bounded by the lines $x=-5$, $x=1$, and the curves $y=10x$ and $y=x^2-11$?

Find the area of the region bounded by the curves $y=6x^2\ln \left(x\right)$ and $y=24\ln \left(x\right)$ for $x>0$.

Find the area of the region enclosed by the inner loop of the curve $r=6+12 \sin \left(\theta \right)$.

Find the area enclosed by one loop of the polar curve $r=4\sin \left(3\theta \right)$.

Which of the following integrals correctly represents the area of the region enclosed by the curves $y=4x$, $y=16x$, and $y=\frac{1}{16}x$ for $x>0$?

What is the area of the region enclosed by the curves $y={\sec }^2\left(x\right)$ and $y=8\cos \left(x\right)$ for $-\frac{\pi }{3}\le x\le \frac{\pi }{3}$?