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 3: Evaluation of Double Integral**
Evaluate the double integral:
\[
\int_{0}^{4} \int_{x}^{2} \sin(y^3) \, dy \, dx
\]
Hint: You may want to reverse the order of integration.
**Explanation for Reversing the Order of Integration:**
The given integral involves integrating with respect to \(y\) first, then \(x\). The bounds for \(y\) are from \(x\) to 2, and for \(x\) are from 0 to 4. To reverse the order of integration, analyze the region over which you are integrating.
**Steps to Reverse the Order:**
1. **Sketch the Region of Integration:**
- The inner integral for \(y\) is from \(y = x\) to \(y = 2\).
- The outer integral for \(x\) is from 0 to 4.
2. **Identify New Bounds:**
- Exchange the roles of \(x\) and \(y\) to find the new limits.
- \(x\) will vary between 0 and \(y\).
- \(y\) will vary from 0 to 2.
3. **Write the Reversed Integral:**
The double integral with the reversed order is:
\[
\int_{0}^{2} \int_{0}^{y} \sin(y^3) \, dx \, dy
\]
This form may be more convenient for evaluation. Integrate first with respect to \(x\) followed by \(y\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd4f38ada-5e94-4e58-9061-25686ec52dc9%2F6bf50fae-691e-4d67-b94c-0fc5e6fba7a4%2Fmhyy1hm_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3: Evaluation of Double Integral**
Evaluate the double integral:
\[
\int_{0}^{4} \int_{x}^{2} \sin(y^3) \, dy \, dx
\]
Hint: You may want to reverse the order of integration.
**Explanation for Reversing the Order of Integration:**
The given integral involves integrating with respect to \(y\) first, then \(x\). The bounds for \(y\) are from \(x\) to 2, and for \(x\) are from 0 to 4. To reverse the order of integration, analyze the region over which you are integrating.
**Steps to Reverse the Order:**
1. **Sketch the Region of Integration:**
- The inner integral for \(y\) is from \(y = x\) to \(y = 2\).
- The outer integral for \(x\) is from 0 to 4.
2. **Identify New Bounds:**
- Exchange the roles of \(x\) and \(y\) to find the new limits.
- \(x\) will vary between 0 and \(y\).
- \(y\) will vary from 0 to 2.
3. **Write the Reversed Integral:**
The double integral with the reversed order is:
\[
\int_{0}^{2} \int_{0}^{y} \sin(y^3) \, dx \, dy
\]
This form may be more convenient for evaluation. Integrate first with respect to \(x\) followed by \(y\).
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