Evaluating a Double Integral by Hand

In the previous videos we looked at a Reimman sum for finding the volume of the function f(x)=24-x^2-3y^2 on the Region R=[0,3]×[0,2], which is a compact way of communicating the region where x is between 0 and 3 and y is between 0 and 2.
We then looked at translating that Reimman sum into an integral

Here we’ll quickly go through the nuts and bolts of evaluating this algebraically, comparing it to what we had in CP3D, and evaluating the integral with MATLAB.
The process is very similar to what you’re familiar with in calc 2, so let’s get started:
Basically, you just work from the inside out to get ‘partial’ integrals while treating one of the variables as a constant.
We will first evaluate the inside integral which is with respect to y,
So we add a y to the 24 constant, add a y to x^2 which we treat as a constant, and use the inverse power rule to integrate 3y^2

Next of course we evaluate by plugging in 2 and 0 for y
V=∫_0^3▒[24(2)-x^2 (2)-(2)^3 ]dx

Now we’ve integrated the inner integral, and next we integrate the outer integral, which is with respect to x this time
So first we take the antiderivatives…
Then we plug in 3 and 0 for x
And performing those operations we arrive at our final answer
V=102
Next we'll verify this in Calcplot 3d and in the next video we'll evaluate in MATLAB anddemonstrate Fubini's theorem.