https://file.aiquickdraw.com/mathbot/file/1726977804367ljk4zaqk.jpg

The problem provided involves finding the volume of a solid generated by revolving the shaded region between the curves \( y = 6x - x^2 \) and \( y = x \) about the **y-axis**.

### Part (a) - Setting up the integral

To find the volume of a solid obtained by revolving a region around the **y-axis**, we use the method of **shell integration**. The formula for the volume of a solid of revolution using cylindrical shells is:

\[
V = 2\pi \int_{a}^{b} x \cdot (f(x) - g(x)) \, dx
\]

Where:
- \( f(x) \) is the upper function (in this case, \( y = 6x - x^2 \))
- \( g(x) \) is the lower function (in this case, \( y = x \))
- \( a \) and \( b \) are the limits of integration (the points where the curves intersect).

#### Step 1: Find the limits of integration

Set the two functions equal to find the points of intersection:

\[
6x - x^2 = x
\]

Rearranging this equation:

\[
x^2 - 5x = 0
\]

Factoring:

\[
x(x - 5) = 0
\]

Thus, \( x = 0 \) and \( x = 5 \) are the points of intersection, so the limits of integration are from \( 0 \) to \( 5 \).

#### Step 2: Set up the volume integral

Using the cylindrical shells method, the volume integral becomes:

\[
V = 2\pi \int_{0}^{5} x \cdot \left[(6x - x^2) - x \right] \, dx
\]

Simplifying the integrand:

\[
(6x - x^2) - x = 5x - x^2
\]

So the integral becomes:

\[
V = 2\pi \int_{0}^{5} x \cdot (5x - x^2) \, dx
\]

### Part (b) - Evaluating the integral

Now we evaluate the integral:

\[
V = 2\pi \int_{0}^{5} \left(5x^2 - x^3\right) \, dx
\]

First, find the antiderivative of the integrand:

\[
\int \left(5x^2 - x^3\right) \, dx = \frac{5x^3}{3} - \frac{x^4}{4}
\]

Now, apply the limits from 0 to 5:

\[
V = 2\pi \left[\left(\frac{5(5)^3}{3} - \frac{(5)^4}{4}\right) - \left(\frac{5(0)^3}{3} - \frac{(0)^4}{4}\right)\right]
\]

Simplify:

\[
V = 2\pi \left[\frac{5(125)}{3} - \frac{625}{4}\right]
\]

\[
V = 2\pi \left[\frac{625}{3} - \frac{625}{4}\right]
\]

Find a common denominator:

\[
V = 2\pi \left[\frac{2500}{12} - \frac{1875}{12}\right]
\]

\[
V = 2\pi \cdot \frac{625}{12}
\]

\[
V = \frac{1250\pi}{12}
\]

\[
V = \frac{625\pi}{6}
\]

Thus, the volume of the solid is:

\[
V = \frac{625\pi}{6} \, \text{cubic units}.
\]

---

Would you like more details on any part of this solution, or do you have any questions?

### 5 Related Questions:
1. How does the method of cylindrical shells differ from the disk/washer method in volume calculation?
2. What are other applications of cylindrical shell integration in real-world problems?
3. How do you determine whether to use cylindrical shells or disks when rotating about different axes?
4. What happens to the volume if the region is rotated about the x-axis instead of the y-axis?
5. How would the integral change if the region were bounded by different functions?

### Tip:
Always ensure the integrand is simplified as much as possible before integrating to avoid unnecessary complications in the evaluation process.

Set up and evaluate an integral to find the volume of a solid formed by revolving the region between the curves y = 6x - x^2 and y = x about the y-axis.

This page explains how to calculate the volume of a solid formed by revolving the region between the curves y = 6x - x^2 and y = x about the y-axis. Learn the step-by-step process of using the cylindrical shell method and solving the integral.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation