https://file.aiquickdraw.com/mathbot/file/17300871276360hveikht.jpg

The problems in the image involve calculating the volume of solids of revolution using the *shell method* ("metode kulit tabung" in Indonesian). Here’s a breakdown of each question and the approach to solve them.

### Problem Breakdown and Solution Approach:

1. **First Problem**:
   - **Region**: Bounded by \( y = \sqrt{x} \), \( x = 4 \), \( x \)-axis, and \( y \)-axis.
   - **Axis of Rotation**: Rotate around \( x = -1 \).
   - **Solution Approach**: Use the shell method to set up the integral, considering the distance of each shell from \( x = -1 \) and the height as given by \( y = \sqrt{x} \).

2. **Second Problem**:
   - **Region**: Bounded by \( y = \sqrt{x} \), \( x = 4 \), \( x \)-axis, and \( y \)-axis.
   - **Axis of Rotation**: Rotate around \( y = 5 \).
   - **Solution Approach**: Again, apply the shell method with the height given by \( y = \sqrt{x} \), but adjust for the rotation about \( y = 5 \) by calculating the distance of the shells accordingly.

3. **Third Problem**:
   - **Region**: The area between \( y = x^2 \) and \( y = \sqrt{8x} \).
   - **Axis of Rotation**: Rotate around \( y = -2 \).
   - **Solution Approach**: Calculate the height of the shell as the difference \( \sqrt{8x} - x^2 \) and use the distance from \( y = -2 \) to determine the radius of each shell.

4. **Fourth Problem**:
   - **Region**: The area between \( y = x^2 \) and \( y = \sqrt{8x} \).
   - **Axis of Rotation**: Rotate around \( x = 3 \).
   - **Solution Approach**: Using the shell method with the rotation around \( x = 3 \), set the height as before and adjust the radius accordingly based on the distance from \( x = 3 \).

Would you like detailed solutions for each problem, including the setup of integrals and calculations? 

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### Related Questions:
1. How do you determine the radius of the shell in each rotation scenario?
2. What adjustments are made to the integral bounds when rotating around different axes?
3. How does the choice of \( x \)- or \( y \)-axis impact the shell method setup?
4. Why might one prefer the shell method over the disk/washer method for these problems?
5. How does changing the axis of rotation to a non-axis line (like \( x = -1 \) or \( y = -2 \)) affect the calculation?

### Tip:
For the shell method, always clearly identify the radius and height of each shell with respect to the chosen axis of rotation before setting up the integral.

Tentukan volume benda putar berikut dengan metode kulit tabung.

Learn how to find the volume of a solid of revolution by using the shell method. The problems include regions bounded by curves like y = sqrt(x) and y = x^2, with rotation around various axes, such as x = -1, y = 5, y = -2, and x = 3.

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