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
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...
Related questions
Question
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce60c797-cede-4a9b-a194-cc1528c2f13e%2F366ed14c-5fa8-4057-89b8-f7b5b2ef6356%2Fq98t2ge_processed.png&w=3840&q=75)
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.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781319050740/9781319050740_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
![Precalculus](https://www.bartleby.com/isbn_cover_images/9780135189405/9780135189405_smallCoverImage.gif)
![Calculus: Early Transcendental Functions](https://www.bartleby.com/isbn_cover_images/9781337552516/9781337552516_smallCoverImage.gif)
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning