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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![### Problem
Sketch the region in the plane over which the integral
\[
\int_{0}^{4} \int_{\sqrt{y}}^{2} \sqrt{1+x^3} \, dx \, dy
\]
is evaluated, then evaluate the integral by reversing the order of integration.
### Graph/Diagram Explanation
The provided diagram is a simple coordinate plane with horizontal and vertical axes (x and y axes, respectively). No specific functions or lines have been drawn on it.
### Steps to Solve:
1. **Understand the Limits of Integration:**
- The outer integral with respect to \( y \) runs from \( 0 \) to \( 4 \).
- The inner integral with respect to \( x \) runs from \( \sqrt{y} \) to \( 2 \).
2. **Sketch the Region:**
- \( y \) ranges from \( 0 \) to \( 4 \).
- For each fixed \( y \), \( x \) ranges from \( \sqrt{y} \) to \( 2 \).
- The line \( x = 2 \) is vertical and constant.
- The curve \( x = \sqrt{y} \) corresponds to \( y = x^2 \), which is a parabola opening upwards.
- Identify the region in the xy-plane bounded by:
- \( y = x^2 \)
- \( x = 2 \)
- \( y = 0 \)
- \( y = 4 \)
3. **Reverse the Order of Integration:**
- Determine the new limits of integration for \( x \) and \( y \).
- The parabola \( y = x^2 \) gives a lower bound for \( y \).
- Reassess the bounds for \( x \) from its minimum value to its maximum value within the region.
4. **Evaluate the Reversed Integral:**
- Perform the integration with the new bounds.
This process will help you evaluate the given integral by reversing the order of integration, allowing for simpler calculations in some cases.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faff7206f-9910-4eba-9e78-ffe8dd60a76e%2Fb61d2d4d-c0f1-4e62-8c3d-70f3bf79c27d%2Fo1d3fe3_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem
Sketch the region in the plane over which the integral
\[
\int_{0}^{4} \int_{\sqrt{y}}^{2} \sqrt{1+x^3} \, dx \, dy
\]
is evaluated, then evaluate the integral by reversing the order of integration.
### Graph/Diagram Explanation
The provided diagram is a simple coordinate plane with horizontal and vertical axes (x and y axes, respectively). No specific functions or lines have been drawn on it.
### Steps to Solve:
1. **Understand the Limits of Integration:**
- The outer integral with respect to \( y \) runs from \( 0 \) to \( 4 \).
- The inner integral with respect to \( x \) runs from \( \sqrt{y} \) to \( 2 \).
2. **Sketch the Region:**
- \( y \) ranges from \( 0 \) to \( 4 \).
- For each fixed \( y \), \( x \) ranges from \( \sqrt{y} \) to \( 2 \).
- The line \( x = 2 \) is vertical and constant.
- The curve \( x = \sqrt{y} \) corresponds to \( y = x^2 \), which is a parabola opening upwards.
- Identify the region in the xy-plane bounded by:
- \( y = x^2 \)
- \( x = 2 \)
- \( y = 0 \)
- \( y = 4 \)
3. **Reverse the Order of Integration:**
- Determine the new limits of integration for \( x \) and \( y \).
- The parabola \( y = x^2 \) gives a lower bound for \( y \).
- Reassess the bounds for \( x \) from its minimum value to its maximum value within the region.
4. **Evaluate the Reversed Integral:**
- Perform the integration with the new bounds.
This process will help you evaluate the given integral by reversing the order of integration, allowing for simpler calculations in some cases.
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