9. Graphical Applications of Integrals

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?

Shade the region bounded by $f\left(x\right)=\sin x$ & $g\left(x\right)=\cos 2x$ on the interval $[\frac{\pi }{6},\frac{5\pi }{6}]$. Then set up an integral to represent the region's area.

Which of the following integrals correctly represents the area of the region enclosed by the curves $y=2x$, $y=8x$, and $y=18x$ for $x>0$?

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 $x>0$.

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 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?

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

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.

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

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

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

41–48. Geometry problems Use a table of integrals to solve the following problems.

46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.

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 $0\le x\le \pi$.