Determine the volume of the solid abtained when the region bounded by y=sqrt x and the line Y=2 and x=0 is rotated:

Question

(i) about the x-axis
(ii) about the line x=4.

(i) Each element of the body of revolution with thickness dx, has area
pi*(sqrt x)^2 dx = pi x dx.
Integrate that from x=0 to x = 4. (The sqrt x curve ends at x=4, y=2.) I get a volume of 8 pi.

(ii) The x=4 vertical line is at the end of the segment of the curve from x=0 to x=4. Perform an integration aloing the y axis. Each slab of thickess dy has volume
pi (4-x)^2 dy. Substitute y^2 for x and the integral becomes
(INTEGRAL sign) pi (4- y^2)^2 dy from y=0 to y=2.