Find the volume obtained by rotating the area

Question

Find the volume obtained by rotating the area bounded by y=x and y= sqrt(x) about the line y=1.‏

Answer 1

Hi Ahmed R.,

A useful simple calculus problem.

Useful steps in solving: try to visualize the generated solid. In this case, since it's a figure rotated around the y=1 line (which lies in the xy plane), the solid is symmetrical in the y and z directions, and has a simple progressive behavior in the x direction. In particular, it's a series of rings: at x = 0, the ring is infinitely thin and radius =1 ; as x increases the ring outer radius decreases but the inner radius decreases even faster, so that the ring gets smaller but thicker; then, as x approaches 1, the ring becomes again infinitely thin with radius = 0 (since the curves cross at (0,0) and (1,1) in the (xy) plane).

Given this behavior, it would be suicidal to try to integrate by dy! but fine to integrate by dx:

V = integral (x=0 to 1) of (x^2 - ((x^(0.5))^2) dx {the two terms in the integrand there are the respective areas of the outer and inner circles bounding the ring slice being integrated} . You should be able to simplify that to integrand (x^2 - x) dx and solve easily, right?

If you get interested in trying to apply your skills to a neat real-world problem, comment back and I'll supply a challenge problem which you could take back to your class (or equivalent, if you're remote at this point!).

-- Cheers, -- Mr. d.