### Finding the Volume of the Given Solid Using Polar CoordinatesTo find the volume of the solid bounded below the plane $6x + y + z = 8$ and above the disk $x^2 + y^2 \leq 1$, we can use polar coordinates for simplification.### Problem StatementWe want to determine the volume of the solid that lies:- Below the plane: $6x + y + z = 8$- Above the disk: $x^2 + y^2 \leq 1$### Key Steps:1. **Convert Cartesian Coordinates to Polar Coordinates:** - $x = r \cos \theta$ - $y = r \sin \theta$ - $r^2 = x^2 + y^2$ 2. **Express the Equation of the Plane in Polar Coordinates:** Given the plane equation $6x + y + z = 8$, substitute $x$ and $y$: \[ 6(r \cos \theta) + r \sin \theta + z = 8 \] \[ z = 8 - 6r \cos \theta - r \sin \theta \]3. **Set Up the Integral:** - The region in the $xy$-plane is a disk with radius 1. - The limits for $r$ are from 0 to 1. - The limits for $\theta$ are from 0 to $2\pi$. The volume integral in polar coordinates becomes: \[ V = \int_{0}^{2\pi} \int_{0}^{1} \left(8 - 6r \cos \theta - r \sin \theta\right) r \, dr \, d\theta \]4. **Evaluate the Integral:** - Integrate with respect to $r$ first. - Then integrate with respect to $\theta$.### Graphical Representation- There is a small rectangular area with a red cross below it, which might be an indicator for students to notice a detail or correction in a given context.By analyzing the components and evaluating the integral correctly, you will find the desired volume of the solid.

According to the given information, it is required to find the volume of the given solid.

Answered: Use polar coordinates to find the…

Use polar coordinates to find the volume of the given solid. Below the plane 6x + y + z = 8 and above the disk x2 + y2 s 1

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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Q: find the volume using polar coordinate system

Question

### Finding the Volume of the Given Solid Using Polar Coordinates

To find the volume of the solid bounded below the plane $6x + y + z = 8$ and above the disk $x^2 + y^2 \leq 1$, we can use polar coordinates for simplification.

### Problem Statement

We want to determine the volume of the solid that lies:
- Below the plane: $6x + y + z = 8$
- Above the disk: $x^2 + y^2 \leq 1$

### Key Steps:

1. **Convert Cartesian Coordinates to Polar Coordinates:**
- $x = r \cos \theta$
- $y = r \sin \theta$
- $r^2 = x^2 + y^2$

2. **Express the Equation of the Plane in Polar Coordinates:**
Given the plane equation $6x + y + z = 8$, substitute $x$ and $y$:
\[
6(r \cos \theta) + r \sin \theta + z = 8
\]
\[
z = 8 - 6r \cos \theta - r \sin \theta
\]

3. **Set Up the Integral:**
- The region in the $xy$-plane is a disk with radius 1.
- The limits for $r$ are from 0 to 1.
- The limits for $\theta$ are from 0 to $2\pi$.

The volume integral in polar coordinates becomes:
\[
V = \int_{0}^{2\pi} \int_{0}^{1} \left(8 - 6r \cos \theta - r \sin \theta\right) r \, dr \, d\theta
\]

4. **Evaluate the Integral:**
- Integrate with respect to $r$ first.
- Then integrate with respect to $\theta$.

### Graphical Representation
- There is a small rectangular area with a red cross below it, which might be an indicator for students to notice a detail or correction in a given context.

By analyzing the components and evaluating the integral correctly, you will find the desired volume of the solid.

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