By changing to polar coordinates, evaluate the integral where D is the disk x² + y² ≤ 36. Answer= (x +ỷ)52 dxdy
By changing to polar coordinates, evaluate the integral where D is the disk x² + y² ≤ 36. Answer= (x +ỷ)52 dxdy
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement:**
By changing to polar coordinates, evaluate the integral
\[
\iint_D (x^2 + y^2)^{5/2} \, dx \, dy
\]
where \( D \) is the disk \( x^2 + y^2 \leq 36 \).
**Solution:**
*Note: To solve this problem, convert the given integral from Cartesian coordinates to polar coordinates.*
**Explanation:**
- The disk \( x^2 + y^2 \leq 36 \) represents a circle with a radius of 6 centered at the origin.
- In polar coordinates, \( x^2 + y^2 = r^2 \) and \( dx \, dy = r \, dr \, d\theta \).
**Convert the Integral:**
The integral becomes:
\[
\int_0^{2\pi} \int_0^6 (r^2)^{5/2} \cdot r \, dr \, d\theta
\]
Simplify the expression:
\[
\int_0^{2\pi} \int_0^6 r^6 \, dr \, d\theta
\]
Now, evaluate the integral in the order of \( dr \) and \( d\theta \):
1. Integrate with respect to \( r \):
\[
\int_0^6 r^6 \, dr = \left[ \frac{r^7}{7} \right]_0^6 = \frac{6^7}{7}
\]
2. Integrate with respect to \( \theta \):
\[
\int_0^{2\pi} \frac{6^7}{7} \, d\theta = \frac{6^7}{7} \cdot \theta \bigg]_0^{2\pi} = \frac{6^7}{7} \cdot 2\pi
\]
3. Compute the final answer:
\[
\frac{6^7 \cdot 2\pi}{7}
\]
Thus, the evaluated integral is:
**Answer =** \(\frac{6^7 \cdot 2\pi}{7}\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18fe6d10-65c6-4dea-a463-83c487832ab0%2F3e8e167c-69e4-4a1e-ac68-9033e46fc88e%2Fdhi4u89_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
By changing to polar coordinates, evaluate the integral
\[
\iint_D (x^2 + y^2)^{5/2} \, dx \, dy
\]
where \( D \) is the disk \( x^2 + y^2 \leq 36 \).
**Solution:**
*Note: To solve this problem, convert the given integral from Cartesian coordinates to polar coordinates.*
**Explanation:**
- The disk \( x^2 + y^2 \leq 36 \) represents a circle with a radius of 6 centered at the origin.
- In polar coordinates, \( x^2 + y^2 = r^2 \) and \( dx \, dy = r \, dr \, d\theta \).
**Convert the Integral:**
The integral becomes:
\[
\int_0^{2\pi} \int_0^6 (r^2)^{5/2} \cdot r \, dr \, d\theta
\]
Simplify the expression:
\[
\int_0^{2\pi} \int_0^6 r^6 \, dr \, d\theta
\]
Now, evaluate the integral in the order of \( dr \) and \( d\theta \):
1. Integrate with respect to \( r \):
\[
\int_0^6 r^6 \, dr = \left[ \frac{r^7}{7} \right]_0^6 = \frac{6^7}{7}
\]
2. Integrate with respect to \( \theta \):
\[
\int_0^{2\pi} \frac{6^7}{7} \, d\theta = \frac{6^7}{7} \cdot \theta \bigg]_0^{2\pi} = \frac{6^7}{7} \cdot 2\pi
\]
3. Compute the final answer:
\[
\frac{6^7 \cdot 2\pi}{7}
\]
Thus, the evaluated integral is:
**Answer =** \(\frac{6^7 \cdot 2\pi}{7}\)
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