Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves x=y^(1/2), x=0, and y=4 about the x axis.

v = ∫[0,4] 2πrh dy

where r=y and h=2-√y
v = ∫[0,4] 2πy(2-√y) dy = 32π/5

check, using discs

v = ∫[0,2] πr^2 dx
where r=y=x^2
v = ∫[0,2] πx^4 dx = 32π/5

To use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves x=y^(1/2), x=0, and y=4 about the x-axis, follow these steps:

1. Sketch the region: Start by sketching the curves x=y^(1/2), x=0, and y=4 on a coordinate plane. This will help you visualize the region you're rotating.

2. Determine the limits of integration: To find the volume, you'll need to integrate from the lowest x-value to the highest x-value of the region. You can find these limits by setting the two curves, x=y^(1/2) and x=0, equal to each other and solving for y. So let's solve for y:

x = y^(1/2)
Square both sides:
x^2 = y

Since y=4 is also a boundary, we need to find the x-value when y=4. Plugging y=4 into the equation x=y^(1/2), we get:

x = 4^(1/2)
x = 2

So the limits of integration are from x = 0 to x = 2.

3. Determine the height of the cylindrical shell: The height of each cylindrical shell will be the difference between the two curves at a given x-value. In this case, the height can be calculated as:

h = 4 - y^(1/2)

4. Determine the radius of the cylindrical shell: The radius of each cylindrical shell will be the x-value. That means the radius is simply:

r = x

5. Write the integral for the volume: The volume of the region can be calculated by integrating the product of the height, the radius, and the differential thickness (dx) of the cylindrical shells. So the integral is:

V = ∫[from 0 to 2] (2πrh) dx

Substituting the expressions for r and h:

V = ∫[from 0 to 2] (2πx(4 - x^(1/2))) dx

Simplify the expression and integrate to find the volume.

I hope this explanation helps you understand how to use the method of cylindrical shells to find the volume!