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Example 5 Video Example Evaluate the following integrals interpreting each in terms of areas. (a) √9-x² dx (b) [(x-2) dx Solution (a) Since f(x)=√√9-x²20, we can interpret this integral as the area under the curve y√9-x² from 0 to 3✔ . But, because y² - 9-x² y=√√9-x² x+y=9 Q Therefore, · S² √9 = x² dx = £27(3)² = [ (b) The graph of y=x-2 is the line with slope 1 ✓shown in the following figure. (7,5) y=x-2 A₁ ↑ We compute the integral as the difference of the areas of the two triangles. (x-2) dx = A₁ - A₂ - [ 1-2-15 X. ,we get x2 + y29, which shows that the graph of fis the quarter-circle with radius 3✔✔ in the figure below.

Example 5 Video Example Evaluate the following integrals interpreting each in terms of areas. (a) √9-x² dx (b) [(x-2) dx Solution (a) Since f(x)=√√9-x²20, we can interpret this integral as the area under the curve y√9-x² from 0 to 3✔ . But, because y² - 9-x² y=√√9-x² x+y=9 Q Therefore, · S² √9 = x² dx = £27(3)² = [ (b) The graph of y=x-2 is the line with slope 1 ✓shown in the following figure. (7,5) y=x-2 A₁ ↑ We compute the integral as the difference of the areas of the two triangles. (x-2) dx = A₁ - A₂ - [ 1-2-15 X. ,we get x2 + y29, which shows that the graph of fis the quarter-circle with radius 3✔✔ in the figure below.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Example 5 Video Example) Evaluate the following integrals by interpreting each in terms of areas. (a) √₁² √9-x²0 dx (b) √² (x - 2) dx Solution (a) Since f(x) = √9 - x² ≥ 0, we can interpret this integral as the area under the curve y = √9 - x² from 0 to 3 ✓ y y=√√9-x² or x² + y² = 9 3 Therefore, · √²³ √9 - x² dx = 1 17(3)² = [ (b) The graph of y = x - 2 is the line with slope 1 shown in the following figure. y (7,5) y = x-2 A1 2 A₂ We compute the integral as the difference of the areas of the two triangles. [²(x-2)0 (x - 2) dx = A₁ - A₂ = - 2 = 1.5 x . . But, because y² = 9-x² , we get x² + y² = 9, which shows that the graph of f is the quarter-circle with radius 3 in the figure below.
Transcribed Image Text:Example 5 Video Example) Evaluate the following integrals by interpreting each in terms of areas. (a) √₁² √9-x²0 dx (b) √² (x - 2) dx Solution (a) Since f(x) = √9 - x² ≥ 0, we can interpret this integral as the area under the curve y = √9 - x² from 0 to 3 ✓ y y=√√9-x² or x² + y² = 9 3 Therefore, · √²³ √9 - x² dx = 1 17(3)² = [ (b) The graph of y = x - 2 is the line with slope 1 shown in the following figure. y (7,5) y = x-2 A1 2 A₂ We compute the integral as the difference of the areas of the two triangles. [²(x-2)0 (x - 2) dx = A₁ - A₂ = - 2 = 1.5 x . . But, because y² = 9-x² , we get x² + y² = 9, which shows that the graph of f is the quarter-circle with radius 3 in the figure below.
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