9. Graphical Applications of Integrals

Centers of Mass and Centroids

Find the centroid of a thin, flat plate covering the region enclosed by the parabolas 𝔂 = 2𝓍² and 𝔂 = 3 ― 𝓍² .

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

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.

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(d) F(4)

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

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

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

7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.

a. lim(x→∞)A(t)

73. Find the area between the curves y=ln(x) and y=ln(2x) from x=1 to x=5.

Consider the region bounded by the graphs of y = sin⁻¹(x), y = 0, and x = 1/2.

b. Find the centroid of the region.

81. Find the lengths of the following curves.

a. y = (x²/8) - ln(x), 4≤x≤8

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

The region in the first quadrant bounded by y=5/2−1/x and y=x

Find the area of the shaded region between $f\left(x\right)=\sin 2x$ & $g\left(x\right)=2\sin x$ from $x=0$ to $x=2\pi$.

Find the area between $f\left(x\right)=x^2-4$ & $g\left(x\right)=-x^2+4$.

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.