Use polar coordinates to find the volume of the given solid. Under the paraboloid z = x2 + y2 and above the disk x2 + y2 < 49
Use polar coordinates to find the volume of the given solid. Under the paraboloid z = x2 + y2 and above the disk x2 + y2 < 49
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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![**Problem Description:**
Use polar coordinates to find the volume of the given solid.
**Given Conditions:**
- The solid is under the paraboloid \( z = x^2 + y^2 \).
- The solid is above the disk \( x^2 + y^2 \leq 49 \).
**Explanation:**
This problem involves calculating the volume of a three-dimensional region. The surface \( z = x^2 + y^2 \) represents a paraboloid opening upwards, while the disk \( x^2 + y^2 \leq 49 \) defines the boundary in the xy-plane with radius 7. The task is to find the volume of the region that lies under the paraboloid and above this disk.
**Approach:**
To solve the problem, you'll use polar coordinates, where \( x = r \cos\theta \) and \( y = r \sin\theta \). This makes it easier to integrate over circular regions:
1. **Convert to Polar Coordinates:**
- Convert the paraboloid's equation to polar form: \( z = r^2 \).
- The limit for \( r \) is from 0 to 7 (since \( r^2 \leq 49 \)).
- The limit for \( \theta \) is from 0 to \( 2\pi \).
2. **Set up the Integral:**
- The volume \( V \) is given by the double integral:
\[
V = \int_{0}^{2\pi} \int_{0}^{7} r^2 \cdot r \, dr \, d\theta
\]
- The \( r \) in \( r^2 \cdot r \) comes from the polar coordinate transformation of the area element \( dx \, dy = r \, dr \, d\theta \).
3. **Evaluate the Integral:**
- First, integrate with respect to \( r \).
- Then, integrate with respect to \( \theta \).
Completing these steps will yield the volume of the solid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5bc766f-fad7-4343-b342-499d4af4816f%2Fdf89c088-137f-4f9d-8a5b-10db80e83c03%2Fcyjo6p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Use polar coordinates to find the volume of the given solid.
**Given Conditions:**
- The solid is under the paraboloid \( z = x^2 + y^2 \).
- The solid is above the disk \( x^2 + y^2 \leq 49 \).
**Explanation:**
This problem involves calculating the volume of a three-dimensional region. The surface \( z = x^2 + y^2 \) represents a paraboloid opening upwards, while the disk \( x^2 + y^2 \leq 49 \) defines the boundary in the xy-plane with radius 7. The task is to find the volume of the region that lies under the paraboloid and above this disk.
**Approach:**
To solve the problem, you'll use polar coordinates, where \( x = r \cos\theta \) and \( y = r \sin\theta \). This makes it easier to integrate over circular regions:
1. **Convert to Polar Coordinates:**
- Convert the paraboloid's equation to polar form: \( z = r^2 \).
- The limit for \( r \) is from 0 to 7 (since \( r^2 \leq 49 \)).
- The limit for \( \theta \) is from 0 to \( 2\pi \).
2. **Set up the Integral:**
- The volume \( V \) is given by the double integral:
\[
V = \int_{0}^{2\pi} \int_{0}^{7} r^2 \cdot r \, dr \, d\theta
\]
- The \( r \) in \( r^2 \cdot r \) comes from the polar coordinate transformation of the area element \( dx \, dy = r \, dr \, d\theta \).
3. **Evaluate the Integral:**
- First, integrate with respect to \( r \).
- Then, integrate with respect to \( \theta \).
Completing these steps will yield the volume of the solid.
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