User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Calculus III Final
I took the final yesterday and I'd like to know whether I got the question correct.

I was given some rectangular coordinates and asked to set up a double integral in polar coordinates to evaluate the volume of the space (I didn't need to actually calculate the integral). It ended up being the volume bounded by the top half of a sphere with radius $8$ and a circle on the $xy$ plane with radius $4$ excluding a cylindrical area in the middle, all centered at the origin. Kind of like half a pitted olive with the pimento taken out. Can anyone tell me how they would set up that integral, to see if I did it correctly? TIA. --GFauxPas 08:44, 10 May 2012 (EDT)


 * Do you mean a triple integral or were you given the formula for a volume of revolution? --Lord_Farin 09:55, 10 May 2012 (EDT)
 * I suspect something like (under the premise that I am imagining the correct volume):
 * $\displaystyle \int_0^{2\pi} \int_4^8 \int_0^{8^2-r^2} r \,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
 * is very close (but this is really cylindrical coordinates). Of course, it trivially reduces to:
 * $\displaystyle \int_4^8\int_0^{8^2-r^2}2\pi r \,\mathrm dz\,\mathrm dr$
 * but this isn't polar coordinates. It does seem like it yields a nice result, though. And indeed, a rather simple calculation yields $1152\pi$. --Lord_Farin 10:06, 10 May 2012 (EDT)