Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = 4. The curves are y = 3x⁴, y = 0, and x = 2.

Answer 1

To find the volume V generated by rotating the region bounded by the curves y = 3x⁴, y = 0, and x = 2 about x = 4, use the method of cylindrical shells.

Step-by-step explanation:

To find the volume V generated by rotating the region bounded by the curves y = 3x⁴, y = 0, and x = 2 about x = 4, we can use the method of cylindrical shells.

1. First, sketch the region bounded by the curves and the line x = 4. This will give you an idea of what the solid shape will look like.
2. Next, determine the height of each cylindrical shell. Since we are rotating about x = 4, the height will be the difference between the x-values of the curves, which is 4 - 2 = 2.
3. Then, find the radius of each cylindrical shell. This will be the y-value of the curve y = 3x⁴ at each x-value. So, the radius is 3(2)⁴ = 48.
4. Finally, use the formula for the volume of a cylindrical shell V = 2πrh, where r is the radius and h is the height, to calculate the volume of each cylindrical shell. Sum up the volumes of all the cylindrical shells to get the total volume V.