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
![**Exercise: Reverse the Order of Integration**
Evaluate the integral by reversing the order of integration.
\[ \int_{0}^{1} \int_{y}^{1} 12e^{-3x^2} \, dx \, dy \]
Options:
- **A.** \( 0 \)
- **B.** \( 0(e^{-6} - 1) \)
- **C.** \( -2(e^{-3} - 1) \)
- **D.** \( 12(e^{-3} - 1) \)
- **E.** \( (e^{-6} - 1) \)
---
**Solution Explanation:**
The initial integral is set up with the limits of integration such that for each fixed \( y \), \( x \) ranges from \( y \) to 1. To reverse the order of integration, visualize or sketch the region of integration in the xy-plane:
1. **Region of Integration:**
- The original limits describe a region where \( 0 \leq y \leq 1 \) and \( y \leq x \leq 1 \).
- When reversing the order, the region should cover \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq x \).
2. **Reversed Integral:**
- Rewriting the integral with these new limits becomes:
\[ \int_{0}^{1} \int_{0}^{x} 12e^{-3x^2} \, dy \, dx \]
3. **Integration Steps:**
- First, integrate the inner integral with respect to \( y \), leading to:
\[ \int_{0}^{x} 12e^{-3x^2} \, dy = [12e^{-3x^2} \cdot y]_{0}^{x} = 12xe^{-3x^2} \]
- Then, perform the outer integration with respect to \( x \):
\[ \int_{0}^{1} 12xe^{-3x^2} \, dx \]
- Use substitution to integrate:
- Let \( u = -3x^2 \), then \( du = -6x \, dx \), so \( -\frac{1}{6} \, du = x \, dx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39ee3059-6733-456c-bbc4-46bedd3c18f2%2F03c2fbf3-f735-46a5-9195-3246ccb56a3e%2Fa0slay_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise: Reverse the Order of Integration**
Evaluate the integral by reversing the order of integration.
\[ \int_{0}^{1} \int_{y}^{1} 12e^{-3x^2} \, dx \, dy \]
Options:
- **A.** \( 0 \)
- **B.** \( 0(e^{-6} - 1) \)
- **C.** \( -2(e^{-3} - 1) \)
- **D.** \( 12(e^{-3} - 1) \)
- **E.** \( (e^{-6} - 1) \)
---
**Solution Explanation:**
The initial integral is set up with the limits of integration such that for each fixed \( y \), \( x \) ranges from \( y \) to 1. To reverse the order of integration, visualize or sketch the region of integration in the xy-plane:
1. **Region of Integration:**
- The original limits describe a region where \( 0 \leq y \leq 1 \) and \( y \leq x \leq 1 \).
- When reversing the order, the region should cover \( 0 \leq x \leq 1 \) and \( 0 \leq y \leq x \).
2. **Reversed Integral:**
- Rewriting the integral with these new limits becomes:
\[ \int_{0}^{1} \int_{0}^{x} 12e^{-3x^2} \, dy \, dx \]
3. **Integration Steps:**
- First, integrate the inner integral with respect to \( y \), leading to:
\[ \int_{0}^{x} 12e^{-3x^2} \, dy = [12e^{-3x^2} \cdot y]_{0}^{x} = 12xe^{-3x^2} \]
- Then, perform the outer integration with respect to \( x \):
\[ \int_{0}^{1} 12xe^{-3x^2} \, dx \]
- Use substitution to integrate:
- Let \( u = -3x^2 \), then \( du = -6x \, dx \), so \( -\frac{1}{6} \, du = x \, dx
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
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