When converted to an iterated integral, the following double integral is easier to evaluate in one order than the other. Find the best order and evaluate the integral. 2x sec? (xy)dA; R = {(x.y): 0 sxs.0sys3)
When converted to an iterated integral, the following double integral is easier to evaluate in one order than the other. Find the best order and evaluate the integral. 2x sec? (xy)dA; R = {(x.y): 0 sxs.0sys3)
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 Resource: Evaluating a Double Integral**
When converted to an iterated integral, the following double integral is easier to evaluate in one order than the other. Find the best order and evaluate the integral:
\[
\iint\limits_{R} 2x \, \sec^2(xy) \, dA, \quad R = \{(x, y) : 0 \leq x \leq \frac{\pi}{9}, \, 0 \leq y \leq 3\}
\]
**Explanation:**
This is a double integral over the region \( R \), defined by the specified limits for \( x \) and \( y \). The function to be integrated is \( 2x \, \sec^2(xy) \), which involves both \( x \) and \( y \).
To solve this, consider reordering the integration based on the given bounds and the function structure. Evaluate the integral by examining variable dependencies, which might simplify calculations.
1. **Integration with Respect to \( y \):** Begin by integrating with respect to \( y \), treating \( x \) as a constant within the inner integral, and then integrate the resulting expression with respect to \( x \).
2. **Integration with Respect to \( x \):** Alternatively, switch the order of integration, starting with \( x \) and then \( y \), based on function or derivative simplicity.
Choosing the correct order can significantly simplify the integration process, especially when functions like \( \sec(xy) \), involving trigonometric identities, are involved.
Graphical or numerical methods may aid in visualizing this region if necessary, assisting in confirming bounds and understanding the function within the defined area.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc986032-179a-454d-ad20-2c76f1bd2607%2F38fc5641-8d5f-481e-88bd-e8648f2d837a%2Fmjcgk8t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Resource: Evaluating a Double Integral**
When converted to an iterated integral, the following double integral is easier to evaluate in one order than the other. Find the best order and evaluate the integral:
\[
\iint\limits_{R} 2x \, \sec^2(xy) \, dA, \quad R = \{(x, y) : 0 \leq x \leq \frac{\pi}{9}, \, 0 \leq y \leq 3\}
\]
**Explanation:**
This is a double integral over the region \( R \), defined by the specified limits for \( x \) and \( y \). The function to be integrated is \( 2x \, \sec^2(xy) \), which involves both \( x \) and \( y \).
To solve this, consider reordering the integration based on the given bounds and the function structure. Evaluate the integral by examining variable dependencies, which might simplify calculations.
1. **Integration with Respect to \( y \):** Begin by integrating with respect to \( y \), treating \( x \) as a constant within the inner integral, and then integrate the resulting expression with respect to \( x \).
2. **Integration with Respect to \( x \):** Alternatively, switch the order of integration, starting with \( x \) and then \( y \), based on function or derivative simplicity.
Choosing the correct order can significantly simplify the integration process, especially when functions like \( \sec(xy) \), involving trigonometric identities, are involved.
Graphical or numerical methods may aid in visualizing this region if necessary, assisting in confirming bounds and understanding the function within the defined area.
Transcribed Image Text:**Problem Statement:**
Find the volume of the given solid over the indicated region of integration.
**Function:**
\( f(x, y) = 8x + 3y + 5 \)
**Region of Integration:**
\( R = \{ (x, y) : -1 \leq x \leq 1, \ 3 \leq y \leq 5 \} \)
**Explanation:**
This problem involves calculating the volume under the surface defined by \( f(x, y) \) over a specified rectangular region \( R \) in the xy-plane. The limits of \( x \) range from \(-1\) to \(1\), and the limits of \( y \) range from \(3\) to \(5\). The integration will provide the volume under the surface above this region.
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