Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Text for Educational Website:**
**Problem Statement:**
Sketch the region of integration and evaluate by changing to polar coordinates:
\[
\int_{0}^{1/2} \int_{\sqrt{3x}}^{\sqrt{1-x^2}} 6x \, dy \, dx = \text{[Insert Solution]}
\]
**Explanation:**
This problem involves calculating a double integral by first sketching the given region of integration in the xy-plane and then converting the Cartesian coordinates to polar coordinates for evaluation.
**Steps to Follow:**
1. **Identify the Region of Integration:**
- The bounds for \(x\) are from 0 to \(\frac{1}{2}\).
- The bounds for \(y\) depend on \(x\), going from \( \sqrt{3x} \) to \( \sqrt{1-x^2} \).
2. **Convert to Polar Coordinates:**
- Use the transformations \(x = r\cos\theta\) and \(y = r\sin\theta\).
- Update the integration bounds and the integrand function to reflect these transformations.
3. **Integrate:**
- Set up the integral in terms of \(r\) and \(\theta\).
- Evaluate the integral to find the solution.
This approach simplifies the evaluation by converting complicated regions in Cartesian coordinates to more manageable regions in polar coordinates.](https://content.bartleby.com/qna-images/question/d3444a3f-7d5c-4474-ab08-7930ed9e066d/b438a640-0b32-4073-acbd-0db92dfa24e6/ks8rdnb_processed.png)
Transcribed Image Text:**Text for Educational Website:**
**Problem Statement:**
Sketch the region of integration and evaluate by changing to polar coordinates:
\[
\int_{0}^{1/2} \int_{\sqrt{3x}}^{\sqrt{1-x^2}} 6x \, dy \, dx = \text{[Insert Solution]}
\]
**Explanation:**
This problem involves calculating a double integral by first sketching the given region of integration in the xy-plane and then converting the Cartesian coordinates to polar coordinates for evaluation.
**Steps to Follow:**
1. **Identify the Region of Integration:**
- The bounds for \(x\) are from 0 to \(\frac{1}{2}\).
- The bounds for \(y\) depend on \(x\), going from \( \sqrt{3x} \) to \( \sqrt{1-x^2} \).
2. **Convert to Polar Coordinates:**
- Use the transformations \(x = r\cos\theta\) and \(y = r\sin\theta\).
- Update the integration bounds and the integrand function to reflect these transformations.
3. **Integrate:**
- Set up the integral in terms of \(r\) and \(\theta\).
- Evaluate the integral to find the solution.
This approach simplifies the evaluation by converting complicated regions in Cartesian coordinates to more manageable regions in polar coordinates.
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