9. Graphical Applications of Integrals

Working with area functions Consider the function ƒ and its graph.

(c) Sketch a graph of A, for 0 ≤ 𝓍 ≤ 10 , without a scale on the y-axis.

72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].

Arc length Use the result of Exercise 108 to find the arc length of the curve: f(x) = ln |tanh(x / 2)| on [ln 2, ln 8].

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.

∫ₐ⁰ ƒ(𝓍) d𝓍

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.

a. Find the area of the region bounded by the graph of f and the x-axis on the interval [0,2].

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.

b. Find the area of the region bounded by the graph of g and the x-axis on the interval [0,2].

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.

c. Which region has greater area?

{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,

∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C

Graph the following functions and find the area under the curve on the given interval.

f(x) = 1/(x√(x² - 36)), [12/√3 , 12]

{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,

∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C

Graph the following functions and find the area under the curve on the given interval.

f(x) = (9 - x²) ⁻², [0, 3/2]

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.

∫₀ᵃ ƒ(𝓍) d𝓍

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.

∫ₐᶜ ƒ(𝓍) d𝓍

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.

∫₀ᶜ |ƒ(𝓍)| d𝓍

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

b. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, 4], where a > 0.

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

c. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, b]. Because this area depends on a and b, we call it A(a, b).

Visual proof Let F(x)=∫₀ˣ √(a²−t²) dt. The figure shows that F(x)= area of sector OAB+ area of triangle OBC.

a. Use the figure to prove that

F(x) = (a² sin ⁻¹(x/a))/2 + x√(a²−x²)/2

b. Conclude that ∫ √(a²−x²) dx = (a² sin ⁻¹(x/a))/2 + x√(a²−x²)/2 + C.