INSTRUCTOR: Calculate the mass, moments, and the center of mass of the region between the curves y equals x and y equals x squared with the density function rho of x comma y equal to x in the interval 0 is less than or equal to x is less than or equal to 1. It's worth noting that in between those x values 0 to 1, the function that is the lower function, has a lower value, is x squared. So this means that our mass, by definition, will be equal to the integral from 0 to 1 with respect to x of the integral from x squared to x with respect to y of the function x, where x is equal to r rho of xy. That's how we find the mass.

Find the antiderivative with respect to y and evaluating, this becomes the integral from 0 to 1 with respect to x of the function x squared minus x cubed, which will then be equal to, again, finding the antiderivative with respect to x and evaluating, this is 1/12. We need to find the mass first because that will play into our moments. And then our moments will play into our center of mass.

So m sub x, the moment in the y direction will be equal to the integral from 0 to 1 with respect to x of the integral from x squared to x with respect to y of the function xy where the moment the with respect to the y-axis is y times rho. Now, finding our antiderivative, this becomes the integral from 0 to 1 with respect to x of one half x cubed minus one half x to the fifth, which finding our antiderivative, evaluating this becomes 1/24. Next, we'll find m sub y, which, applying our definition again, this is the integral from 0 to 1 with respect to x of the integral from x squared to x with respect to y of the function x squared.

Evaluating the inner integral, this becomes the integral from 0 to 1 with respect to x of x to the third, or x cubed minus x to the fourth, which is equal to 1 over 20. Now, our center of mass is found by dividing, in fact, that I'll right this. It's x bar comma y bar as a point. This will be equal to m sub y divided by our mass m comma m sub x divided by our mass m. So if we divide those values, our center of mass is 3/5 comma one half.