4. For Parts (a) (b), sketch the region of integration and evaluate the integral by hand with work shown. Please circle your final answer. •3y Згу dx dy (») LL 4y dy dx
4. For Parts (a) (b), sketch the region of integration and evaluate the integral by hand with work shown. Please circle your final answer. •3y Згу dx dy (») LL 4y dy dx
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 4:
For Parts (a) and (b), sketch the region of integration and evaluate the integral by hand with work shown. Please circle your final answer.
#### (a)
\[
\int_{0}^{2} \int_{y^2}^{3y} 3xy \, dx \, dy
\]
#### (b)
\[
\int_{-1}^{1} \int_{0}^{\sqrt{4-x^2}} 4y \, dy \, dx
\]
### Explanation
For both parts, you need to:
1. **Sketch the Region of Integration:**
- **Part (a):**
- The region is bounded by \(y = x^2\) and \(x = 3y\) from \(y = 0\) to \(y = 2\).
- **Part (b):**
- The region is bounded by \(y = 0\) to \(y = \sqrt{4-x^2}\) (upper half of a circle with radius 2 centered at the origin) from \(x = -1\) to \(x = 1\).
2. **Evaluate the Integral:**
- Show the step-by-step calculation for each part.
- Circle the final numerical answer.
This exercise helps reinforce the concept of evaluating double integrals by hand and understanding the region of integration for given limits.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4229329e-d022-49ed-b0e8-7dc5623f42be%2Fe7ad4c13-0956-40c5-9cbf-c20e7d68e14a%2Fyt9dvur_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 4:
For Parts (a) and (b), sketch the region of integration and evaluate the integral by hand with work shown. Please circle your final answer.
#### (a)
\[
\int_{0}^{2} \int_{y^2}^{3y} 3xy \, dx \, dy
\]
#### (b)
\[
\int_{-1}^{1} \int_{0}^{\sqrt{4-x^2}} 4y \, dy \, dx
\]
### Explanation
For both parts, you need to:
1. **Sketch the Region of Integration:**
- **Part (a):**
- The region is bounded by \(y = x^2\) and \(x = 3y\) from \(y = 0\) to \(y = 2\).
- **Part (b):**
- The region is bounded by \(y = 0\) to \(y = \sqrt{4-x^2}\) (upper half of a circle with radius 2 centered at the origin) from \(x = -1\) to \(x = 1\).
2. **Evaluate the Integral:**
- Show the step-by-step calculation for each part.
- Circle the final numerical answer.
This exercise helps reinforce the concept of evaluating double integrals by hand and understanding the region of integration for given limits.
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