se a double integral in polar coordinates to find the volume V of the solid bounded by the graphs of the equations. (Round your answer to two decimal places.) z- In(x + y) 2+ 29 2 +ys 25 dr de = leed Help? Read It Master It Talk to a Tutor 10:32 AM W 10/31/2020
se a double integral in polar coordinates to find the volume V of the solid bounded by the graphs of the equations. (Round your answer to two decimal places.) z- In(x + y) 2+ 29 2 +ys 25 dr de = leed Help? Read It Master It Talk to a Tutor 10:32 AM W 10/31/2020
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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![The task is to use a double integral in polar coordinates to find the volume \( V \) of the solid bounded by the graphs of the equations. Round your answer to two decimal places.
Equations given:
\[ z = \ln(x^2 + y^2) \]
\[ z = 0 \]
\[ x^2 + y^2 > 9 \]
\[ x^2 + y^2 \leq 25 \]
The setup for the integral in polar coordinates is provided:
\[ V = \int_{0}^{2\pi} \int_{3}^{5} (\ln(r^2)) \, r \, dr \, d\theta \]
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**Explanation of the Task:**
To solve the problem, we need to evaluate a double integral in polar coordinates to find the volume of a cylinder-like solid defined by the given equations. Here is a breakdown of the setup:
1. **Region of Integration:**
- The region is defined in polar coordinates where \( r \) ranges from 3 to 5, which corresponds to the circular annulus defined by \( 9 < x^2 + y^2 \leq 25 \).
- The angle \( \theta \) ranges from \( 0 \) to \( 2\pi \), covering the entire circle.
2. **Function to be Integrated:**
- The function we are integrating is \( \ln(r^2) \times r \). Here, \( \ln(r^2) \) is derived from converting \( \ln(x^2 + y^2) \) to polar coordinates. The extra \( r \) comes from the Jacobian determinant when converting from Cartesian to polar coordinates.
3. **Integration Limits:**
- The bounds for \( r \) are 3 to 5, as mentioned, and for \( \theta \), it is \( 0 \) to \( 2\pi \).
Once the integral is evaluated, it will give the volume of the solid described by the given equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F6f76584c-f40a-4e5d-9ece-dfb837aac8e0%2F9093e2t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The task is to use a double integral in polar coordinates to find the volume \( V \) of the solid bounded by the graphs of the equations. Round your answer to two decimal places.
Equations given:
\[ z = \ln(x^2 + y^2) \]
\[ z = 0 \]
\[ x^2 + y^2 > 9 \]
\[ x^2 + y^2 \leq 25 \]
The setup for the integral in polar coordinates is provided:
\[ V = \int_{0}^{2\pi} \int_{3}^{5} (\ln(r^2)) \, r \, dr \, d\theta \]
Buttons labeled for help:
- Read It
- Master It
- Talk to a Tutor
**Explanation of the Task:**
To solve the problem, we need to evaluate a double integral in polar coordinates to find the volume of a cylinder-like solid defined by the given equations. Here is a breakdown of the setup:
1. **Region of Integration:**
- The region is defined in polar coordinates where \( r \) ranges from 3 to 5, which corresponds to the circular annulus defined by \( 9 < x^2 + y^2 \leq 25 \).
- The angle \( \theta \) ranges from \( 0 \) to \( 2\pi \), covering the entire circle.
2. **Function to be Integrated:**
- The function we are integrating is \( \ln(r^2) \times r \). Here, \( \ln(r^2) \) is derived from converting \( \ln(x^2 + y^2) \) to polar coordinates. The extra \( r \) comes from the Jacobian determinant when converting from Cartesian to polar coordinates.
3. **Integration Limits:**
- The bounds for \( r \) are 3 to 5, as mentioned, and for \( \theta \), it is \( 0 \) to \( 2\pi \).
Once the integral is evaluated, it will give the volume of the solid described by the given equations.
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