In Problems 41–56, find the area of the region bounded by the graphs of the given equations. Be sure to find any needed points of intersection. Consider whether the use of horizontal strips makes the integral simpler than when vertical strips are used. 41. y=x², y= 2r 42. y = x, y = -x+3, y=0
In Problems 41–56, find the area of the region bounded by the graphs of the given equations. Be sure to find any needed points of intersection. Consider whether the use of horizontal strips makes the integral simpler than when vertical strips are used. 41. y=x², y= 2r 42. y = x, y = -x+3, y=0
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
41
![# Integration
## 38. Refer to Figure 14.40
Figure 14.40 shows a diagram with the curves:
- \( y = x^2 - 5 \)
- \( y = 7 - 2x^2 \)
The region in question is to the right of the line \( x = 1 \).
## 39.
Express, in terms of a single integral, the total area of the region to the right of the line \( x = 1 \) that is between the curves \( y = x^2 - 5 \) and \( y = 7 - 2x^2 \). Do not evaluate the integral.
## 40.
Express, in terms of a single integral, the total area of the region in the first quadrant bounded by the x-axis and the graphs of \( y = x^2 \) and \( y = 3 - x \). Do not evaluate the integral.
## 41-56.
Find the area of the region bounded by the graphs of the given equations. Use \( x = \) or \( y = \) as needed. Consider horizontal strips for simplicity.
- Example: \( y = x^2 + x - 3 \), \( y = x + 3 \), \( y = 0 \)
- Note different forms such as: \( x = y^2 - 4y + 4 \) and similar variations for different equations.
## 59. Lorenz Curve
A Lorenz curve demonstrates income distribution inequality.
- Let \( x \) be the cumulative percentage of income recipients
- Let \( y \) be the cumulative percentage of income.
When equality is achieved, the Lorenz curve is \( y = x \). For example, when \( x = 10\%\), then \( y = 10\%\).
### Figure 14.41
It shows the Lorenz curve defined by:
\[ y = \frac{14}{15}x^2 + \frac{1}{15}x \]
- The graph plots cumulative percentages of income against cumulative percentages of recipients.
- Note: 30% of people receive only 10.4% of total income, indicating inequality.
The degree of deviation from equality is measured by the coefficient of inequality.
## 60-68. Exercises
Determine areas using coordinates, curves, lines, or given inequalities. Calculate the coefficient of inequality for specific Lorenz](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e2d6f28-e3d7-414c-b1d4-a752131175c2%2Fe989b092-8e99-492b-b2cb-8caabbd79123%2Fa62cp8xi.jpeg&w=3840&q=75)
Transcribed Image Text:# Integration
## 38. Refer to Figure 14.40
Figure 14.40 shows a diagram with the curves:
- \( y = x^2 - 5 \)
- \( y = 7 - 2x^2 \)
The region in question is to the right of the line \( x = 1 \).
## 39.
Express, in terms of a single integral, the total area of the region to the right of the line \( x = 1 \) that is between the curves \( y = x^2 - 5 \) and \( y = 7 - 2x^2 \). Do not evaluate the integral.
## 40.
Express, in terms of a single integral, the total area of the region in the first quadrant bounded by the x-axis and the graphs of \( y = x^2 \) and \( y = 3 - x \). Do not evaluate the integral.
## 41-56.
Find the area of the region bounded by the graphs of the given equations. Use \( x = \) or \( y = \) as needed. Consider horizontal strips for simplicity.
- Example: \( y = x^2 + x - 3 \), \( y = x + 3 \), \( y = 0 \)
- Note different forms such as: \( x = y^2 - 4y + 4 \) and similar variations for different equations.
## 59. Lorenz Curve
A Lorenz curve demonstrates income distribution inequality.
- Let \( x \) be the cumulative percentage of income recipients
- Let \( y \) be the cumulative percentage of income.
When equality is achieved, the Lorenz curve is \( y = x \). For example, when \( x = 10\%\), then \( y = 10\%\).
### Figure 14.41
It shows the Lorenz curve defined by:
\[ y = \frac{14}{15}x^2 + \frac{1}{15}x \]
- The graph plots cumulative percentages of income against cumulative percentages of recipients.
- Note: 30% of people receive only 10.4% of total income, indicating inequality.
The degree of deviation from equality is measured by the coefficient of inequality.
## 60-68. Exercises
Determine areas using coordinates, curves, lines, or given inequalities. Calculate the coefficient of inequality for specific Lorenz
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning