9. Graphical Applications of Integrals

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

ƒ(x) = 5 - 5x²/³, -1 ≤ x ≤ 8

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

y = x, y = 1/x², x = 2

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

√x + √y = 1, x = 0, y = 0

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

x = 2y², x = 0, y = 3

Find the extreme values of ƒ(x) = x³ - 3x², and find the area of the region enclosed by the graph of ƒ and the x-axis.

Find the total area of the region enclosed by the curve x = y²/³ and the lines x = y and y = -1.

Find the lengths of the curves in Exercises 19–22.

y = x¹/² ― (1/3) x³/² , 1 ≤ x ≤ 4

Find the lengths of the curves in Exercises 19–22.

y = (5/12) x⁶/⁵ ― (5/8)x⁴/⁵ , 1 ≤ x ≤ 32

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 .

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

81. Find the lengths of the following curves.

a. y = (x²/8) - ln(x), 4≤x≤8

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.

b. Find the center of mass if, instead of being constant, the density function is δ(x)=4/√x.

In Exercises 139–142, find the length of each curve.

139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.

In Exercises 139–142, find the length of each curve.

141. y = ln(cos(x)) from x = 0 to x = π/4.