Calculus triple integral

Question

Find the mass of the solid bounded below by the circular paraboloid  and above by the circular paraboloid  if the density .Write your answer in terms of .

Answer 1

Using Cylindrical Coordinates.

The projection of the intersection of the two circular paraboloids on the xy-plane is the circle x² + y² = 13/8

Then the mass of the solid is equal

m= ∫_θ=0 ^θ=2π ∫_r=0 ^{r= √(13/8)} ∫_z=r^2 ^{z= 13/4-r^2} r² dz dr dθ =

m = ∫_θ=0 ^θ=2π ∫_r=0 ^{r= √(13/8)}[ r²z ] |_r^2^{13/4-r^2} dr dθ =

m = ∫_θ=0 ^θ=2π ∫_r=0 ^{r= √(13/8)}[ r² [ 13/4 -2r² ] dr dθ =

m = ∫_θ=0 ^θ=2π ∫_r=0 ^{r= √(13/8)}[ r² 13/4 -2r^4 ] dr dθ =

m = ∫_θ=0 ^θ=2π [ 13r³ /12 - 2r⁵/5] |₀√(13/8) dθ =

m = ∫_θ=0 ^θ=2π [ 13/12 13/8 √(13/8)- 169 /160 √(13/8)dθ

m = 238π( 1/96- 1/160) √(13/8)