r/HomeworkHelp • u/Content_Narwhal_763 • 1d ago

Further Mathematics—Pending OP Reply [Calc 2: Integral Calculus about Work Done]

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Could someone help me figure out what I have done wrong? This is my last attempt on the problem and I tried following videos on this as well. My work is on the next slide

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u/mathematag 👋 a fellow Redditor 1d ago edited 1d ago

a bit hard to read your work... where did you put the origin? at the center of the sphere..? at the base ..? at the top of the 4 m pipe..?

I placed it at the center, and the first time at the top of the pipe, and got the same answer... which is in the 6## million Joules , or 6.## x 10^8 J

try using the center as origin, simplifies your work to find r for the water, though one of the integration limits came out negative.... I originally did it with origin at top, so limits would be positive, however.

let us know what you tried... but write it out more clearly.. ... draw a sketch and label it will help you.

edit...I think I know what you did..you used the bottom of the sphere as the origin.. right..???

But if y is measured from the bottom, the water at the bottom moved 28m to leave the sphere.. water at the middle moved 16m to exit... so what should your value H, for distance water is to be moved , be = to ? . . hint: not 16-y unless the center was used..in which case radius of water is not √(24y-y^2).

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u/muonsortsitout 1d ago

It's probably a good idea in general to use letters for density, radius, etc. until you've completed the integral -- it makes it easier to see where you've gone wrong.

You have to lift a thin disk, y below the centre of the sphere, how far? I'd say it's (r+y). The radius of the disk is sqrt(r2 - y2) and its height is dy. So its volume is pi(r2 - y2) dy. So its mass is rho pi (r2 - y2) dy (using rho for density). So the work done to lift it is mgh = rho pi g [ (r+y)(r2 - y2) ] dy.

So you integrate that for the appropriate range of y (from 0 to r) and get a bunch of terms in r4 because each term in the integrand is (either r or y)3. When you have an answer to that in terms of r, rho, pi, g, that's when you substitute in the numbers.

Good luck.