Evaluating a double integral and demonstrating Fubini's Theorem with MATLAB

In the last video we quickly go through the nuts and bolts of evaluating this algebraically, comparing it to what we had in CP3D, and here we'll evaluated the integral with MATLAB and use MATLAB to demonstrate Fubini's Theorem
First we define our x and y symbolic variables
syms x y
Then we define our function
f(x,y)=24-x^2-3*y^2;
And now we use the integrate command nested within another integrate command,
So our inner integral is wrt y between 0 and 2
And our outer integral is wrt x between 0 and 3
int(int(f,y,[0 2]),x,[0 3])
And there we have our answer of 23 confirmed
Finally, we can also compare this to our Reimann sum from CP3D
And if we go back to that and ratchet up our increments in the x and y directions we see that our Reimann sum gets pretty close to 102, further confirming our answer.
But one final question here, what if we switch around the order of integration, will that give us a different answer?
Well, I would encourage you to do just using the calculus steps above, but since hopefully you trust MATLAB now that we’ve confirmed our answer, let’s try it there..
And… we see this does in fact give us the same answer,
This illustrates ‘Fubini’s Theorem’
So we can freely switch our order of integration as long as we keep the bounds consistent and as long as those bounds are just numbers. In the next video, we’ll discuss what happens when those bounds are actually functions rather than numbers, but until then, take care