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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![**Educational Content: Changing the Order of Integration**
In the given problem, we need to change the order of integration for the double integral:
\[
\int_{0}^{2} \int_{y^2}^{2y} f(x, y) \, dx \, dy
\]
The task is to determine the correct limits of integration when reversing the order from \(dx \, dy\) to \(dy \, dx\).
### Explanation:
The original limits indicate:
- \( y \) ranges from \( 0 \) to \( 2 \).
- For a fixed \( y \), \( x \) ranges from \( y^2 \) to \( 2y \).
To find the new limits:
1. Sketch or visualize the region described by these limits in the xy-plane.
2. Determine the region bounded by solving the equations \( x = y^2 \) and \( x = 2y \).
3. The intersection points of these curves occur where \( y^2 = 2y \), solving this gives \( y(y-2) = 0 \), hence \( y = 0 \) or \( y = 2 \).
Once the sketch is complete, you can derive:
- \( x \) ranges from \( 0 \) to \( 4 \) (since \( x = y^2 \) starts at the origin and extends to \( x = 4 \) where \( y = 2 \)).
- For a fixed \( x \), calculate the limits on \( y \) by solving the expressions for \( y \): \( y = \sqrt{x} \) (from \( x = y^2 \)) and \( y = x/2 \) (from \( x = 2y \)).
The integral with reversed order will thus be set up with these conditions. Select the correct answer that matches these new derived limits.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7417eb4-c4f6-4b23-b299-1f887648c283%2Ffe1d26e7-75a8-4952-8c31-baee4ddece1c%2Fbcdwca_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Content: Changing the Order of Integration**
In the given problem, we need to change the order of integration for the double integral:
\[
\int_{0}^{2} \int_{y^2}^{2y} f(x, y) \, dx \, dy
\]
The task is to determine the correct limits of integration when reversing the order from \(dx \, dy\) to \(dy \, dx\).
### Explanation:
The original limits indicate:
- \( y \) ranges from \( 0 \) to \( 2 \).
- For a fixed \( y \), \( x \) ranges from \( y^2 \) to \( 2y \).
To find the new limits:
1. Sketch or visualize the region described by these limits in the xy-plane.
2. Determine the region bounded by solving the equations \( x = y^2 \) and \( x = 2y \).
3. The intersection points of these curves occur where \( y^2 = 2y \), solving this gives \( y(y-2) = 0 \), hence \( y = 0 \) or \( y = 2 \).
Once the sketch is complete, you can derive:
- \( x \) ranges from \( 0 \) to \( 4 \) (since \( x = y^2 \) starts at the origin and extends to \( x = 4 \) where \( y = 2 \)).
- For a fixed \( x \), calculate the limits on \( y \) by solving the expressions for \( y \): \( y = \sqrt{x} \) (from \( x = y^2 \)) and \( y = x/2 \) (from \( x = 2y \)).
The integral with reversed order will thus be set up with these conditions. Select the correct answer that matches these new derived limits.
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