9. Graphical Applications of Integrals

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

The region in the first quadrant bounded by y=x^2/3 and y=4

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.

Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.

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

The region bounded by x=y(y−1) and y=x/3

106. Arc length Find the length of the curve y = (x / 2) * sqrt(3 - x^2) + (3 / 2) * sin^(-1)(x / sqrt(3)) from x = 0 to x = 1.

Find the area of the region bounded by the curves $y=\sin 2x$ and $y=\sin 3x$ for $0\le x\le \pi$.

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.

√x + √y = 1, x = 0, y = 0

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

42. Find the length of the curve y = x^(3/2) + 8 on the interval from 0 to 2.

Area of a sector of a hyperbola: Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus

a. What is the area of R?

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

123. Region between curves Find the area of the region bounded by the graphs of y = tan(x) and y = sec(x) on the interval [0, π/4].

Find the area of the region (see figure) in two ways.

a. Using integration with respect to x.

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.                                                                                                                                      

 ∫₀⁴ √(16― 𝓍² ) d𝓍

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(a) A(2)

Find the shaded area between $f\left(x\right)=x^3+2x^2$ & $g\left(x\right)=x+2$.