An elliptical galaxy has gravitational boundaries defiend by 9x^{2}+16y^{2}+144z^{2}=144. A black hole at the center of the galaxy is interacting with dark matter producing a radiation envelope defined by the region inside the cone Z^{2}=27x^{2}+48y^{2}. If the energy density in this envelope is defined by ρ(x,y,z)=Κ |xyz|
Determine the following given the above with K >C
1. Find the volume of space inside the galaxy affected by the interaction using a triple integral transformed into u,v,w space with u=3x, v=4y, w=z
2.the total energy in the envelope.
3. the center of "mass of the energy in the upper and lower cones individually in the (x,y,z) system.
4.the average energy density , ρ, in each cone.
Note: for units you may use the following. linear dimensions = 10^5 light year