Find mass of solid: 9x^2+z^2=y^2, y=9, above z=0

[MarkSA]

Hi,

1) Find the mass of the solid bounded by elliptic cone 9x^2 + z^2 = y^2, and the plane y=9, and above the plane z=0, if the density is equal to the distance from the xy-plane. Do this by integrating the "inside" integral with respect to "y".

This is how i've set it up. As I understand the sketch, i'm looking at the upper half of 2 elliptic cones, one opening along the positive y and the other opening along the negative y. One is cut by plane y=9 and the other by y=-9. I am actually not positive this is correct, but it was the only way I could get the correct answers to work out for the first two parts of the question, which were to do the same procedure by integrating the inside integral with respect to x and y.

\(\displaystyle \int^{9}_{0}\int^{\sqrt{\frac{81-z^2}{9}}}_{-\sqrt{\frac{81-z^2}{9}}}\int^{\sqrt{9x^2+z^2}}_{-\sqrt{9x^2+z^2}} z \ dydxdz\)

The correct answer should be 729/2, but i'm pretty far off from that. I'm really stumped on this one.

 

[galactus]

Re: Find the mass of the solid

There are always many ways to set this up, provided we set them up correctly. We can use symmetry and multiply by 2:

Since \(\displaystyle 9x^{2}=81, \;\ x=3\) by setting z=0 and y=9.

So, we get:

\(\displaystyle 2\int_{0}^{3}\int_{0}^{\sqrt{81-9x^{2}}}\int_{\sqrt{9x^{2}+z^{2}}}^{9}z \;\ dydzdx=\frac{729}{2}\)

 

[MarkSA]

Re: Find the mass of the solid

Argggh.. so apparently there is just one cone on this problem. So that would make my above limits wrong.

So I went through and tried it again (I know it can be done the way you showed but I really want to figure out why this way doesn't work. If I did this on a test i'd think I had the right answer only to end up with a big fat 0.)

\(\displaystyle \int^{9}_{0}\int^{\sqrt{\frac{81-z^2}{9}}}_{-\sqrt{\frac{81-z^2}{9}}}\int^{\sqrt{9x^2+z^2}}_{0} z \ dydxdz\)

Why on earth does this not work? The y bounds go from 0 to the end of the cone, the x bounds cover both sides of the cone from the negative to positive, and the z-bounds should be right. Evaluated, I get an answer of 2187/2 which is 3 * 729/2.
Ugh, i've spent all weekend studying these problems and I feel like I haven't made any progress understanding how to do them.