Two-Dimensional Integrals Using the Monte Carlo Method
Two-Dimensional Integrals Using the Monte Carlo Method
Consider the family of surfaces defined by , where . This Demonstration plots the surface and approximates the two-dimensional integral zdxdy, the volume under the surface, using a Monte Carlo approximation method. You can vary the values of the parameters , , and and the number of randomly generated points on the surface. The approximate volume, given by , is compared to the result from Mathematica's built-in function NIntegrate.
z=f(x,y)=4+α-β+2γxy
2
x
2
y
-1≤x,y≤1
1
∫
-1
1
∫
-1
α
β
γ
N
N
{(,)}
x
i
y
i
i=1
V=f(,)
4
N
N
∑
i=1
x
i
y
i