9. Graphical Applications of Integrals

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

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.

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

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 $-\frac{\pi }{3}\le x\le \frac{\pi }{3}$?

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 $x>0$?

Determine the area of the shaded region in the following figures.

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

Find the area of the region R bounded by the graphs of $y=x^2+1$, $y=2x$, and $x=0$.

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

Find the area of the region described in the following exercises.

The region bounded by y=e^x, y=2e^−x+1, and x=0

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.

Calculate the area of the shaded region between the 2 functions from $x=0$ to $x=9$

Calculate the area of the shaded region between $f\left(x\right)$ & $g\left(x\right)$ contained between $x=-4$ & $x=-2$.

Sketch the region bounded by $f\left(x\right)=-{(x-2)}^2+5$ & $g\left(x\right)=4x$ on the interval $[0,1]$. Then set up an integral to represent the region's area and evaluate.