Question

How do I find the regions for the triple integral to calculate the mass of the the volume between the paraboloid x^2+y^2 = 2az and the sphere x^2+y^2+z^2=3a^2
with a density of d(x,y,z)=x^2+y^2+z^2?

Answer 1

is that similar to finding the solution to a system of equations in order to compute that volume of a solid?

Answer 2

I didn't quite understand what you mean...

To be more clear, I must find out the volume between the two objects, then calculate it's mass. (oh and a>0)

Answer 3

oh... so the sphere and the paraboloid arent intersecting are they...lol. Wrong concept on my part :)

Answer 4

yes they are intersecting, and I must find that volume.

Answer 5

when two curves meet and we want to find the area between them, we integrate the bounds between where they meet; is that something we can do to the sphere and parabaloid?

Answer 6

when does:

x^2 +y^2 -2az = x^2 +y^2 +z^2-3az^2 ?

Answer 7

it intersects in a circle in the plane z=a

Answer 8

that circle is one bound of the interecting shapes then; do we have an x bound and a ybound yet?

Answer 9

we are trying, if i see it correctly, find the area that is scooped out of an ice cream container :)

Answer 10

area means volume lol

Answer 11

there should be an upper and lower bound of all three dimensions

Answer 12

if we could extablish a cross section; could we spin it to find the volume of the solid created by the rotation about an axis?

Answer 13

Unusually complicated. I would try\[\int\limits_{0}^{2\pi}\int\limits_{r ^{2}/2a}^{\sqrt{3a ^{2}-r ^{2}}}\int\limits_{a \sqrt{2(3a ^{2}-1)}}^{a ^{2\sqrt{6}}}r ^{3+z ^{2}drdzd \theta}\]

Answer 14

thanks guys, I'll give it a try. Although I'm not so sure about the limits of dr