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
![## Switch the Order of Integration
Switch the **order of integration** of the following integrals, then **evaluate**:
\[ \int_{x=0}^{1} \int_{y=x}^{1} x \sin(y^3) \, dy \, dx = \]
### Steps to switch the order of integration:
1. **Identify the original limits:**
- For \( x \) from 0 to 1.
- For \( y \) from \( x \) to 1.
2. **Graphical representation:**
The graph depicts the region of integration in the \( xy \)-plane, represented by a triangular region bounded as follows:
- The x-axis, from 0 to 1.
- The line \( y = x \).
- The vertical line \( y = 1 \).
3. **Rewriting the integral with switched order:**
To switch the order of integration, express the region in terms of \( y \) first and then \( x \):
- \( y \): from 0 to 1
- \( x \): from 0 to \( y \)
Thus, the integral becomes:
\[ \int_{y=0}^{1} \int_{x=0}^{y} x \sin(y^3) \, dx \, dy \]
### Evaluate the integral after switching
1. **Inner integral evaluation:**
\[\int_{x=0}^{y} x \sin(y^3) \, dx\]
\( x \sin(y^3) \) is treated as a constant with respect to \( x \):
\[ \sin(y^3) \int_{x=0}^{y} x \, dx = \sin(y^3) \left[ \frac{x^2}{2} \right]_{0}^{y} = \sin(y^3) \cdot \frac{y^2}{2} \]
2. **Result of the inner integral:**
\[ \frac{y^2}{2} \sin(y^3) \]
3. **Outer integral evaluation:**
\[\int_{y=0}^{1} \frac{y^2}{2} \sin(y^3) \, dy\]
- Let \( u = y^3 \), hence \( du = 3y^2 \,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b50af26-7d61-4a7c-9b13-b01063ee7292%2Fcaa8ea57-d492-426f-8b7b-d6866378301c%2Fi9s2emk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Switch the Order of Integration
Switch the **order of integration** of the following integrals, then **evaluate**:
\[ \int_{x=0}^{1} \int_{y=x}^{1} x \sin(y^3) \, dy \, dx = \]
### Steps to switch the order of integration:
1. **Identify the original limits:**
- For \( x \) from 0 to 1.
- For \( y \) from \( x \) to 1.
2. **Graphical representation:**
The graph depicts the region of integration in the \( xy \)-plane, represented by a triangular region bounded as follows:
- The x-axis, from 0 to 1.
- The line \( y = x \).
- The vertical line \( y = 1 \).
3. **Rewriting the integral with switched order:**
To switch the order of integration, express the region in terms of \( y \) first and then \( x \):
- \( y \): from 0 to 1
- \( x \): from 0 to \( y \)
Thus, the integral becomes:
\[ \int_{y=0}^{1} \int_{x=0}^{y} x \sin(y^3) \, dx \, dy \]
### Evaluate the integral after switching
1. **Inner integral evaluation:**
\[\int_{x=0}^{y} x \sin(y^3) \, dx\]
\( x \sin(y^3) \) is treated as a constant with respect to \( x \):
\[ \sin(y^3) \int_{x=0}^{y} x \, dx = \sin(y^3) \left[ \frac{x^2}{2} \right]_{0}^{y} = \sin(y^3) \cdot \frac{y^2}{2} \]
2. **Result of the inner integral:**
\[ \frac{y^2}{2} \sin(y^3) \]
3. **Outer integral evaluation:**
\[\int_{y=0}^{1} \frac{y^2}{2} \sin(y^3) \, dy\]
- Let \( u = y^3 \), hence \( du = 3y^2 \,
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 2 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