Monte Carlo Volume Technique

The Python solution involves generating random points (x,y,z) in the TargetBox with 
 a resulting  Plane-z (0<= z <=3). Plane-z intersects the volume in a Circle and Hyperbola  
(picture above). If x and y lie in the circle/hyperbola area then the point (x,y,z)  lies in the 
volume between the two surfaces.

Import numpy as np
#                                           **Make Use Of The Above 3-D graph**
# TargetBoxVol=3*(2.75)**2; If plane z=3,3=(2/5)*(x**2+y**2) then max radius=sqrt(7.5)=2.74
def fCir(y,z):
      return np.sqrt(abs(2.5*z-y**2))
def fHyp(y,z):
      return np.sqrt(abs(1*z+y**2))
# FindHitsInHP counts the number of times points (x,y,z) lies between the two surfaces    
def FindHitsInHP(x,y,z): #x y z are column arrays  
      x,y=np.abs(x),np.abs(y)
      yR=np.sqrt(1.5*z) # y coordinate of intersection of Circle and Hyperbola
      yLR=(y<=yR) #Is y an element of the interval [0,yR]?
      #Is x in the interval[fHyp(y,z),fCir(y,z)]?
      trueHP=((fHyp(y,z)<=x)&(x<=fCir(y,z))) 
      return(sum(trueHP&yLR)) #sums all Trues in the array trueHP
#End FindHitsInHP           
#Main
seed=9007
#np.random.seed(seed) Use as needed for testing
Trials,SSize,TotVolHP=20,500000,0
TarS=2.75 #Target Side Length
TarH=3 #Target Height
TargetBoxVol=TarH*(2*TarS)**2  
for i in range(1,Trials+1):
    x,y,z=np.random.uniform(-TarS,TarS,SSize),np.random.uniform(-TarS,TarS,SSize),np.random.uniform(0,3,SSize)
    HitsInHP=FindHitsInHP(x,y,z)
    VolHP=(HitsInHP/SSize)*TargetBoxVol
    print('Total (x,y,z) in Vol',HitsInHP,'-Volume Common to Hyperbolic-Paraboloid and a Paraboloid',VolHP,SSize,'Trial',i)
    TotVolHP+=VolHP
print('Average Volume ',TotVolHP/Trials)
Total (x,y,z) in Vol 330348-Volume Common to (Hyperbolic-Paraboloid and a Paraboloid) 5.9958 5000000 Trial 1
Total (x,y,z) in Vol 330412-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9969 5000000 Trial 2
Total (x,y,z) in Vol 330122-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9917 5000000 Trial 3
Total (x,y,z) in Vol 330261-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9942 5000000 Trial 4
Total (x,y,z) in Vol 33063-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 6.0010 5000000 Trial 5
Total (x,y,z) in Vol 330157-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9923 5000000 Trial 6
Total (x,y,z) in Vol 330180-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9927 5000000 Trial 7
Total (x,y,z) in Vol 329652-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9831 5000000 Trial 8
Total (x,y,z) in Vol 330650-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 6.0012 5000000 Trial 9
Total (x,y,z) in Vol 329596-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9822 5000000 Trial 10
Total (x,y,z) in Vol 329703-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9841 5000000 Trial 11
Total (x,y,z) in Vol 329939-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9883 5000000 Trial 12
Total (x,y,z) in Vol 330375-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9963 5000000 Trial 13
Total (x,y,z) in Vol 330260-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9941 5000000 Trial 14
Total (x,y,z) in Vol 330405-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9964 5000000 Trial 15
Total (x,y,z) in Vol 329373-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9781 5000000 Trial 16
Total (x,y,z) in Vol 330626-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 6.0008 5000000 Trial 17
Total (x,y,z) in Vol 329731-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9846 5000000 Trial 18
Total (x,y,z) in Vol 330303-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9949 5000000 Trial 19
Total (x,y,z) in Vol 329548-Volume Common to Hyperbolic-Paraboloid and a Paraboloid 5.9812 5000000 Trial 20
**Average Volume  5.99156**