9. Graphical Applications of Integrals

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

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

y² = 4x, y = 4x - 2

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

y = sin x, y = x, 0 ≤ x ≤ π/4

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

y = 2 sin x, y = sin 2x, 0 ≤ x ≤ π

Find the area of the “triangular” region bounded on the left by x + y = 2, on the right by y = x², and above by y = 2.

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 .

Centers of Mass and Centroids

Find the center of mass of a thin, flat plate covering the region enclosed by the parabola 𝔂² = 𝓍 and the line 𝓍 = 2𝔂 if the density function is δ(𝔂) = 1 + 𝔂. (Use horizontal strips.)