Show that total mass of half solid hemisphere radius a given as x^2 + y^2 + z^2 = a^2 and z>=0 and having variable density ρ=1+(r^2)z where r is the distance of any point from the origin is given by: M = (2π(a^3))(6+3(a^3)/18 where M is total mass.
The Attempt at a Solution
My first attempt was to find the volume of the hemisphere using spherical polar coordinates and multiply this volume with the density given. Obviously this was wrong. I then tried integrating again but this time i integrated 1+(x^2 + y^2 + z^2)z
I replaced the r^2 in the density equation with a^2. I still ended up with the wrong answer. Have i applied the right method the second time round and maybe just made a mistake when integrating, or am i approaching this question the wrong way? Thanks.