Question

Find the volume of the solids generated by revolving the region bounded by

y = √x, y = 2, x = 0
about the following axes. You can use disks/washers or shells, but remember that drawing a good picture of each case is the key.
(a) about the y-axis
(b) about the line y = 2

Answer 1

To find the volume of the solid generated by revolving the region bounded by y = √x, y = 2, and x = 0 about the y-axis, we can use the method of disks/washers and integrate the area of each circular cross-section of the solid. The final volume is 32π/3.

Step-by-step explanation:

Solution for (a)

To find the volume of the solid generated by revolving the region bounded by y = √x, y = 2, and x = 0 about the y-axis, we can use the method of disks/washers. We need to integrate the area of each disk or washer, which is a circular cross-section of the solid. The radius of each disk/washer is the value of x, and the height is the difference between y = 2 and y = √x.

We can express the volume as V = π∫(2^2 - (√x)^2) dx, where the limits of integration are x = 0 and x = 4. Solving the integral, we get V = π(4x - (x^(3/2))/2)|[0,4] = π(16 - 8/3) = 32π/3.