9. Graphical Applications of Integrals

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

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?

27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.

Find the area of each of the regions R₁,R₂, and R₃.

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 $x>0$?

Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.

Area

In Exercises 11–14, find the total area of the region between the graph of ƒ and the x-axis.

ƒ(x) = x² - 4x + 3, 0 ≤ x ≤ 3

122. Comparing areas The region R₁ is bounded by the graph of y = tan(x) and the x-axis on the interval [0, π/3].

The region R₂ is bounded by the graph of y = sec(x) and the x-axis on the interval [0, π/6]. Which region has the greater area?

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?

84.a. Find the center of mass of a thin plate of constant density covering the region between the curve y=1/√x and the x-axis from x=1 to x=16.

Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.                                              

 ƒ(𝓍) = 16―𝓍² on [―4, 4]

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = ln (x−√x²−1), for 1 ≤ x ≤ √2(Hint: Integrate with respect to y.)

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.

f(x) = c * x * √(25 - x²) over [0, 5]

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?