https://file.aiquickdraw.com/mathbot/file/1725487080211njraj44m.jpg

The problem in the image involves finding the volume of a solid formed by rotating a region bounded by the curves around the y-axis. The functions and boundaries given are:

- \( y = 2x \)
- \( y = \frac{1}{2} x^2 \)
- \( x = 0 \)
- \( y = 0 \)

Let's proceed with solving this problem using the method of cylindrical shells or disks/washers, whichever is more appropriate for the setup. The goal is to calculate the volume of the solid obtained by rotating the area between these curves about the y-axis.

Would you like a detailed step-by-step solution for this problem? Here are some related questions:

1. How do you determine the bounds of integration when rotating around an axis?
2. What is the method of cylindrical shells, and how is it applied here?
3. How do you set up the volume integral using the washer method for this problem?
4. How do you determine where the curves intersect to find the limits of integration?
5. What is the difference between using the disk method and the shell method for this type of problem?

**Tip:** When setting up volume problems, always carefully sketch the region and visualize the rotation to choose the best method (disks, washers, or shells).

Find the volume of the solid formed by rotating the region bounded by y = 2x, y = (1/2)x^2, x = 0, and y = 0 about the y-axis.

Explore how to find the volume of a solid of revolution by rotating the region bounded by y = 2x, y = (1/2)x^2, x = 0, and y = 0 about the y-axis. This problem covers key calculus concepts such as the shell and washer methods, ideal for advanced calculus students.

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