**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.

Using polar coordinates dV=rdrdθdz and 0≤z≤r2, 0≤r2≤49 or 0≤r≤7.

Answered: Use polar coordinates to find the…

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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Question

**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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