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 $y=x^2$ and $y=2x$ for $x\in [0,2]$?

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 $x>0$?

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

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 enclosed by one loop of the curve $r=\sin \left(6\theta \right)$.

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.