Use Monte Carlo method to estimate the volume of a sphere with radius equal to 3

Question

Use Monte Carlo method to estimate the volume of a sphere with
radius equal to 3 meters (origin at 0,0,0). Use enough data points such that the estimate is
accurate to the 6th decimal digit. Verify your answer using the equation for the volume of a
sphere:


.
Note: In geometry, the equation for a sphere (origin at 0,0,0) in Cartesian coordinates is
+ + =
Hint: To estimate the volume of a sphere, you will need to first estimate the area of the circles
created by the slices created by fixing the z-axis at various values, summing up the areas of all
circles to obtain the sphere’s volume. The interval between successive z values will determine
the number of slices, affecting the accuracy. Of course, the number of samples used in the Monte
Carlo method will also affect the accuracy.

Explanation / Answer

% "Monte Carlo" estimation of the volume of a sphere. Npoints = 1000; radius = 1; x = rand(1, Npoints) - radius; y = rand(1, Npoints) - radius; z = rand(1, Npoints) - radius; distance = sqrt(x.^2 + y.^2 + z.^2); Vcube = (2 * radius)^3; Vsphere = sum(distance

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