Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about the​ y-axis.

y equals 5 x minus x squaredy=5x−x2​, 

yequals=0

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55

y equals 5 x minus x squaredy=5x−x2

xx

yy

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Part 1

Set up the integral that gives the volume of the solid using the shell method. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice.

​(Type exact​ answers.)

A.Integral from nothing to nothing left parenthesis nothing right parenthesis dy

Integral from nothing to nothing left parenthesis nothing right parenthesis dy∫enter your response hereenter your response hereenter your response here dy

B.Integral from 0 to 5 left parenthesis 2 pi x left parenthesis 5 x minus x squared right parenthesis right parenthesis dx

Integral from 0 to 5 left parenthesis 2 pi x left parenthesis 5 x minus x squared right parenthesis right parenthesis dx∫00552 pi x left parenthesis 5 x minus x squared right parenthesis2πx5x−x2 dx

Your answer is correct.

Part 2

The volume is 

enter your response here. 

​(Type an exact​ answer.)

To find the volume of the solid generated by revolving the region \( R \) bounded by the curves \( y = 5x - x^2 \) and \( y = 0 \) around the y-axis using the shell method, we follow these steps:

### Part 1: Set up the integral

1. **Identify the region and the curves:**
   - The region \( R \) is bounded by \( y = 5x - x^2 \) and \( y = 0 \).
   - We need to find the interval over which \( x \) varies. Solve \( y = 0 \) for \( x \):
     \[
     5x - x^2 = 0 \implies x(5 - x) = 0 \implies x = 0 \text{ or } x = 5.
     \]
   - Therefore, \( x \) varies from 0 to 5.

2. **Set up the shell method formula:**
   - For the shell method, the volume \( V \) is given by:
     \[
     V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dx
     \]
   - Here, the radius of each cylindrical shell is the distance from the y-axis, which is \( x \).
   - The height of each shell is the function value \( y \), which is \( 5x - x^2 \).

3. **Write the integral:**
   - The volume integral becomes:
     \[
     V = 2\pi \int_{0}^{5} x (5x - x^2) \, dx
     \]

So, the correct setup for the integral is:
\[
V = 2\pi \int_{0}^{5} x (5x - x^2) \, dx
\]

### Part 2: Evaluate the integral

1. **Simplify the integrand:**
   \[
   V = 2\pi \int_{0}^{5} (5x^2 - x^3) \, dx
   \]

2. **Integrate term by term:**
   \[
   \int_{0}^{5} 5x^2 \, dx = 5 \int_{0}^{5} x^2 \, dx = 5 \left[ \frac{x^3}{3} \right]_{0}^{5} = 5 \left( \frac{5^3}{3} - 0 \right) = 5 \left( \frac{125}{3} \right) = \frac{625}{3}
   \]
   \[
   \int_{0}^{5} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{0}^{5} = \frac{5^4}{4} - 0 = \frac{625}{4}
   \]

3. **Combine the results:**
   \[
   V = 2\pi \left( \frac{625}{3} - \frac{625}{4} \right)
   \]

4. **Find a common denominator and simplify:**
   \[
   V = 2\pi \left( \frac{2500}{12} - \frac{1875}{12} \right) = 2\pi \left( \frac{625}{12} \right) = \frac{625\pi}{6}
   \]

Therefore, the volume of the solid is:
\[
V = \frac{625\pi}{6}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you find the volume of a solid using the washer method?
2. What are the differences between the shell method and the washer method?
3. Can you use the shell method to find the volume of solids of revolution around the x-axis?
4. How do you determine the height of the shell in the shell method?
5. What are common mistakes to avoid when using the shell method?
6. How does the shell method change if the region is revolved around a vertical line other than the y-axis?
7. How do you set up the integral for the washer method?
8. Can the shell method be used for regions bounded by more complicated functions?

### Tip:
When setting up the integral for the shell method, ensure that the radius and height of the shells are correctly identified based on the axis of rotation. This will help avoid common mistakes and ensure the integral is set up correctly.

Learn how to find the volume of a solid generated by revolving a region bounded by the curves y = 5x - x^2 and y = 0 around the y-axis using the shell method. Detailed step-by-step solution provided.

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