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
100%
![**Calculating the Double Integral**
In this section, we will compute a given double integral over a defined region. The double integral in question is:
\[ \iint_R 11x \sin(x + y) \, dA \]
Where the region \( R \) is defined as:
\[ R = \left[0, \frac{\pi}{6}\right] \times \left[0, \frac{\pi}{3}\right] \]
This indicates that our region \( R \) is a rectangle in the \( xy \)-plane where \( x \) ranges from 0 to \( \frac{\pi}{6} \) and \( y \) ranges from 0 to \( \frac{\pi}{3} \).
To evaluate this double integral, we will integrate over \( x \) and \( y \) within the given limits. Here's the detailed breakdown of the region \( R \) and how the bounds are set for both \( x \) and \( y \):
1. **Integral Bounds for \( x \)**: \( 0 \leq x \leq \frac{\pi}{6} \)
2. **Integral Bounds for \( y \)**: \( 0 \leq y \leq \frac{\pi}{3} \)
The integral can be expressed and computed as:
\[ \int_{0}^{\frac{\pi}{3}} \int_{0}^{\frac{\pi}{6}} 11x \sin(x + y) \, dx \, dy \]
This format makes it evident that we will first integrate with respect to \( x \) from 0 to \( \frac{\pi}{6} \), and then integrate with respect to \( y \) from 0 to \( \frac{\pi}{3} \).
The rectangular bounds have been clearly specified to simplify the computation of the double integral, ensuring that all variables are confined within the given intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97b7422a-52f7-4bc4-acdc-28b7d7bc43e4%2F34135f64-f59a-4d91-9e8b-c8d3509d1fc7%2Fdnv0exv_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculating the Double Integral**
In this section, we will compute a given double integral over a defined region. The double integral in question is:
\[ \iint_R 11x \sin(x + y) \, dA \]
Where the region \( R \) is defined as:
\[ R = \left[0, \frac{\pi}{6}\right] \times \left[0, \frac{\pi}{3}\right] \]
This indicates that our region \( R \) is a rectangle in the \( xy \)-plane where \( x \) ranges from 0 to \( \frac{\pi}{6} \) and \( y \) ranges from 0 to \( \frac{\pi}{3} \).
To evaluate this double integral, we will integrate over \( x \) and \( y \) within the given limits. Here's the detailed breakdown of the region \( R \) and how the bounds are set for both \( x \) and \( y \):
1. **Integral Bounds for \( x \)**: \( 0 \leq x \leq \frac{\pi}{6} \)
2. **Integral Bounds for \( y \)**: \( 0 \leq y \leq \frac{\pi}{3} \)
The integral can be expressed and computed as:
\[ \int_{0}^{\frac{\pi}{3}} \int_{0}^{\frac{\pi}{6}} 11x \sin(x + y) \, dx \, dy \]
This format makes it evident that we will first integrate with respect to \( x \) from 0 to \( \frac{\pi}{6} \), and then integrate with respect to \( y \) from 0 to \( \frac{\pi}{3} \).
The rectangular bounds have been clearly specified to simplify the computation of the double integral, ensuring that all variables are confined within the given intervals.
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 3 steps with 3 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