Consider the graph of y^2 = x(4-x)^2 (see link). Find the volumes of the solids that are generated when the loop of this graph is revolved about (a) the x-axis, (b) the y-axis, and (c) the line x = 4.

Question

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I'm just having trouble trying to set up the problem for finding the definite integral to compute the volume. But once I have it set up, I can find the solution pretty easily.

Now for b, since it's revolving about the y-axis, the definite integral setup was given as V = 4pi ∫[0,4] x(4-x)sqrt(x) dx. How did they get that formula exactly and I thought it was supposed to be 2pi not 4pi?
I think I understand how they got the "(4-x)sqrt(x)" part since when solving for y by itself, y = +- sqrt(x)(4-x), but where did that extra x in that definite integral come from???

Answer 1

(a) using discs of thickness dx,
v = ∫ πr^2 dx
where r = y = x(4-x)^2
v = ∫[0,4] π(x(4-x)^2) dx

(b) using shells of thickness dx, and taking advantage of symmetry,

v = 2∫[0,4] 2πrh dx
where r=x and h=y=(4-x)√x
v = 2∫[0,4] 2πx(4-x)√x dx

Answer 2

Thanks!

Would we want to use shells of thickness dx and symmetry for (c) as well?

Answer 3

nope - washers, since it's revolving around a horizontal axis, just as in (a).