2. Find the shaded area shown in the figure first by integrating with respect to x and next by integrating with respect to y. Follow all the steps from the class notes. Which method is easier? 2
2. Find the shaded area shown in the figure first by integrating with respect to x and next by integrating with respect to y. Follow all the steps from the class notes. Which method is easier? 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Y = sqrt of x
Y= (1/2) x
![### Topic: Calculating the Area of a Region Bounded by Curves
#### Exercise Description
**Problem 2**: Find the shaded area shown in the figure first by integrating with respect to \( x \) and next by integrating with respect to \( y \). Follow all the steps from the class notes. Which method is easier?
#### Detailed Explanation
The provided graph displays two curves that form a bounded region with a shaded area. The curves are:
1. \( y = x \) in red.
2. \( y = \sqrt{x} \) in purple.
These curves intersect at the points \( (0,0) \) and \( (1,1) \), giving us the limits of integration for \( x \) from 0 to 1 and for \( y \) from 0 to 1.
##### Integrating with Respect to \( x \)
To calculate the shaded area by integrating with respect to \( x \), we use the expression for the area under a curve:
\[ \text{Area} = \int_{a}^{b} [f(x) - g(x)] \, dx \]
Where \( f(x) \) is the upper curve and \( g(x) \) is the lower curve.
Here,
- The upper curve: \( y = x \)
- The lower curve: \( y = \sqrt{x} \)
The shaded area \( A \) is:
\[ A = \int_{0}^{1} (x - \sqrt{x}) \, dx \]
##### Integrating with Respect to \( y \)
To integrate with respect to \( y \), we use the inverse functions for the integration:
\[ \text{Area} = \int_{c}^{d} [f(y) - g(y)] \, dy \]
Where \( f(y) \) is the right-most curve and \( g(y) \) is the left-most curve.
For this problem,
- The right-most curve: \( x = y^2 \)
- The left-most curve: \( x = y \)
The shaded area \( A \) is:
\[ A = \int_{0}^{1} (y - y^2) \, dy \]
##### Visual Representation
- The intersection points \( (0,0) \) and \( (1,1) \) are clearly marked.
- The area between these curves is shaded, indicating the region of interest.
#####](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff548c102-08e9-4b6b-8c12-787edeafe577%2F76cbcf5e-976d-4599-8edb-a8a71efcd1fc%2F5vcr08h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Topic: Calculating the Area of a Region Bounded by Curves
#### Exercise Description
**Problem 2**: Find the shaded area shown in the figure first by integrating with respect to \( x \) and next by integrating with respect to \( y \). Follow all the steps from the class notes. Which method is easier?
#### Detailed Explanation
The provided graph displays two curves that form a bounded region with a shaded area. The curves are:
1. \( y = x \) in red.
2. \( y = \sqrt{x} \) in purple.
These curves intersect at the points \( (0,0) \) and \( (1,1) \), giving us the limits of integration for \( x \) from 0 to 1 and for \( y \) from 0 to 1.
##### Integrating with Respect to \( x \)
To calculate the shaded area by integrating with respect to \( x \), we use the expression for the area under a curve:
\[ \text{Area} = \int_{a}^{b} [f(x) - g(x)] \, dx \]
Where \( f(x) \) is the upper curve and \( g(x) \) is the lower curve.
Here,
- The upper curve: \( y = x \)
- The lower curve: \( y = \sqrt{x} \)
The shaded area \( A \) is:
\[ A = \int_{0}^{1} (x - \sqrt{x}) \, dx \]
##### Integrating with Respect to \( y \)
To integrate with respect to \( y \), we use the inverse functions for the integration:
\[ \text{Area} = \int_{c}^{d} [f(y) - g(y)] \, dy \]
Where \( f(y) \) is the right-most curve and \( g(y) \) is the left-most curve.
For this problem,
- The right-most curve: \( x = y^2 \)
- The left-most curve: \( x = y \)
The shaded area \( A \) is:
\[ A = \int_{0}^{1} (y - y^2) \, dy \]
##### Visual Representation
- The intersection points \( (0,0) \) and \( (1,1) \) are clearly marked.
- The area between these curves is shaded, indicating the region of interest.
#####
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