INSTRUCTOR: Suppose Q Is a solid region bounded by the plain x plus 2y plus 3z equals 6 and the coordinate planes with density RHO of xyz equal to x times y squared times z. Find the center of mass. The most important thing we need to notice is what the boundaries are going to be in each of these directions, in the z direction, the y direction, and the x direction.

Now, if we notice, because we are bounded by the coordinate planes and this plane, the lower bound is going to be 0 for all of these. So z is bounded from 0 to some function. Y will be bounded from 0 to some function. And x will be bounded actually between two constants.

So first, I'm going to solve this plane, z equals 1/3 open parentheses 6 minus 4x minus 2y. And then if we restrict, so that's our boundary actually on the z, in the z direction. Now, if we restrict ourselves into the xy plane, this now just becomes x plus 2y equals 6, that line. And so we can solve that for y to find the boundary of y. And that would be 1/2 open parentheses, 6 minus x, close parentheses.

Now, these are relationships. So I'm actually going to, just to describe this function, I want to just redefine these so I don't have to rewrite these over and over again. So I'm going to say w is the function 1/3, parentheses 6 minus 4x minus 2y. And I'll use v for my 1/2 parentheses 6 minus x, close parenthesis.

So when I set up my integration, I'm going to use these variables to represent those quantities, just so that I don't have to rewrite that function over and over again. Now, in finding mass, that would be equal to the integral. Oh, I didn't find my x values.

Well, jumping back over here, if we restrict ourselves only onto the x-axis, that means the y and z are both 0, and we find that x goes from 0 to 6. So the mass will be the integral from 0 to 6 with respect to x of the integral from 0 to y. Well, I'm going to call that v, 0 to v with respect to y of the integral from 0 to w with respect to z, thinking about my boundaries, how I've defined those. And that's of the function xy cubed, or y squared, xy squared z. That's our function.

Now, there's a lot of algebra I'm going to skip as we go through here because these are things we should be able to do. So the next step here will be equal to, once we find the antiderivative with respect to z and evaluate our end points, this becomes the integral from 0 to 6 with respect to x of the integral from 0 to v with respect to y of the function 1/18 multiply by xy squared, open parentheses 6 minus 4x minus 2y, close parentheses, squared. Now, find the antiderivative of that function, that inner integral, with respect to y, and evaluating that as well, this becomes the integral from 0 to 6 with respect to x of the function negative x to the sixth divided by 4,320, end fraction, plus x to the fifth over 144, end fraction, minus 1/12 x to the fourth plus 1/2 times x cubed minus 3/2 times x squared plus 9/5 times x.

And we find that integral. That will be 54 over 35. Now, we need that value is the mass for each of the moments or to find the center of mass once we find the moments, that is.

All right, so just scroll down a bit. In the moment in the xy plane, Mxy will be equal to all those same boundaries, the integral from 0 to 6 with respect to x of the integral from 0 to v with respect to y of the integral from 0 to w with respect to z of the function z times x squared, or back up. My function is, in fact, z xy squared z.

All right, now, finding our antiderivative with the innermost integral, this becomes the integral from 0 to 6 with respect to x of the integral from 0 to v with respect to y of the function negative 1/81, end fraction, xy squared, open parenthesis x plus 2y minus 6, close parentheses, cubed. Now, finding the antiderivative of that function and evaluating, we find the integral from 0 to 6 with respect to x of fraction x to the seventh with the denominator of 38,880, end fraction, minus x to the sixth over 1,080, end fraction, plus x to the fifth over 72, end fraction, minus x to the fourth over 9, end fraction, plus x cubed over 2, end fraction, minus 6x squared, denominator of 5, end fraction, plus 6x over 5. And if we evaluate that integral, that will be 27/35.

So there's Mx, or Mxy. Now, we have two more moments to find. That will be in Mxz, which is the moment in the xz plane. That will be the integral from 0 to 6 with respect to x of the integral from 0 to v with respect to y of the integral from 0 to w with respect to z of the function y times xy squared times z.

Following a lot of those same steps, we find this is the integral from 0 to 6 with respect to x of the integral from 0 to v with respect to y of the function 1/18, end fraction, xy cubed, open parenthesis, x plus 2y minus 6, close parentheses, squared, which would then be equal to the integral from 0 to 6 with respect to x of the function x to the seventh over 17,280, end fraction, minus x to the sixth over 480, end fraction, plus x to the fifth, numerator, denominator of 32, minus x to the fourth power with a denominator of 4, plus numerator 9x cubed, denominator of 8, end fraction, minus 27x squared as the numerator with a denominator of 10, end fraction, plus 27x as a numerator with a 10, as a denominator.

That's all with respect to x. Now, we can find that value. That turns out to be 243/140.

Finally, we can find Myz, the moment in the yz plane. That will be equal to the integral from 0 to 6 with respect to x of the integral from 0 to v with respect to y of the integral from 0 to w with respect to z of the function x times xy squared z. Evaluating the inner integral, this becomes the integral from 0 to 6 with respect to x of the integral from 0 to v with respect to y of the function 1/18, end fraction, x squared, y squared, open parenthesis, x plus 2y minus 6, close parentheses, squared.

Evaluating the next integral, this becomes the integral from 0 to 6 with respect to x of negative x to the seventh. That's over 4,320, end fraction, plus x to the sixth over 144, end fraction, minus x to the fifth over 12, end fraction, plus x to the fourth over 2, end fraction, minus 3x cubed over 2, end fraction, plus 9x squared over 5, end fraction. And we can find that value to be 81/35.

Each of these steps is a Calculus 1 step. But, of course, we're doing it with several variables. That's where the skipping some of the algebra and things comes in. A lot of this can turn out to be a bunch of power rules.

OK, now we know that the center of mass, parentheses x bar comma y bar comma z bar, is equal to the point, the ordered triple, which is a point. And its Myz over the mass, M comma, Mxz over the mass M, comma Mxy over the mass M. So dividing each of those values that we've already found, the center of mass is the 0.3 halves comma 9/8, comma 1/2.