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 x comma y comma z equal to xy squared z. Find the moments of inertia about the three coordinate planes. Now, as I noted in the last video, in the last example, we can restrict our functions down to-- The lower limits are all going to be a limit of 0 because we're bounded by the court and planes. And the plane itself will be our upper limit.

Now, in each stage here, we want to think about, if we're living in the z plane, or if we're only living in the xy plane, or just in the x direction, because that's how I'm going to pare this down. So I'm going to say that z is equal to 1/3, open parentheses 6 minus 4x minus 2y. And to represent that quantity, I'm actually going to use the letter w in my limits of integration so that I don't have just keep saying that function over and over again.

Now, if we restrict ourselves out of the z dimension and only think in terms of x and y, then the boundary becomes y equals 1/2 open parentheses 6 minus x, close parentheses, which I'm going to use v to refer to that quantity there. Now, if as we're moving down in dimension, because I'm going from z to y to x, as I move down in dimension, currently, I'm only thinking of a line, when I say y equals. That's a line in the xy plane.

Now, if I go one further, I'm talking about boundaries on the x-axis, which is going to be from 0 to 6. So when I set these up, I'm going to use 0 to 6, my limits for x, 0 to w for my boundaries on z, and 0 to v for my limits on w. Now, let's use our definitions and set up these moments.

So Ix will be equal to 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 parentheses y squared plus z squared, close parentheses, multiplied by xy squared z. Right now, determine the inner interval. This will become 0, integral from 0 to 6 with respect to x of the integral from 0 to v with respect to y with a function 1/324, end fraction, xy squared, open parenthesis, x plus 2y minus 6, close parentheses, squared open parentheses, open parentheses again, x plus 2y minus 6, close parentheses squared plus 18y squared, close parentheses.

Now, I'm skipping a lot of algebra here. So I suspect that you should be able to convince yourself of every line that I have here by doing some of that algebra on your own. After all, when it comes to simplifying, I often will use a CAS, computer algebra system, or do the stuff out by hand separately.

Now this becomes the integral from 0 to 1. And this is very much expanded, with respect to x of the function negative enumerator 11x to the eighth, divided by our denominator of 544,320, end fraction, plus 11x to the seventh as our numerator, with a denominator of 12,960, end fraction, minus 11x to the sixth as our numerator, divided by a denominator of 720, end fraction, plus 11x to the fifth as a numerator, divided by 72, end fraction, minus 11x to the fourth as our numerator. That's divided by 12 in our denominator, end fraction, plus 33x cubed. And that's over 10, end fraction, minus 33x squared, the denominator of 5, dividing by 5 there, end fraction, plus 198 x. And that is divided by 35.

And when we evaluate that integral, that will be the fraction 99/35. And that is our moment about the x. Or that's the moment of inertia. And let's see, that would be with respect to the yz plane.

All right, next we have Iy, this moment. This will be set up very similarly. 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 open parenthesis x squared plus z squared, close parentheses, times xy squared z, which is our row function.

All right, and that would 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 function 1/324, end fraction, xy squared, open parenthesis x plus 2y minus 6, close parentheses squared, open parentheses 18x squared plus open parenthesis x plus 2y minus 6, close parentheses squared, and then close parentheses. Evaluating this integral, this becomes integral from 0 to 6 with respect to x of the function negative 2x to the eighth over 8,505, end fraction, plus 23x to the 7 over 3,240, end fraction, minus 31 x to the sixth over 360, end fraction, plus 19x to the fifth over 36, end fraction, minus 5x to the fourth over 3, end fraction, plus 12x cubed over 5, end fraction, minus 6x squared over 5, end fraction, plus 36x over 35. And that integral, using our power rules and all that, becomes 36/7. That is Iy.

Finally, we'll find Iz. 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 open parenthesis x squared plus y squared, close parentheses, times xy squared z. This 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 function 1 over 18, end fraction, xy squared, open parenthesis x plus 2y minus 6, close parentheses squared, open parenthesis x squared plus y squared.

This will be equal to the integral from 0 to 6 with respect to x of the function negative x to the eighth over 4,032, end fraction, plus 11x to the 7 over 1,440, end fraction, minus 23x to the sixth over 240, end fraction, plus 5x to the fifth over 8, end fraction, minus 9x to the fourth over 4, end fraction, plus 9x cubed over 2, end fraction, minus 27x squared over 5, end fraction, plus 162x over 35. And the value of that integral is 243/35. And that is Iz.

So we have Ix is equal with 99/35. Iy is equal to 36/7. And Iz is 243/35.