00:01
We want to find the center of mass in the mass for an object that occupies the region d, where d is a region of the xy plane whose x values are between 1 and 7, and y values are between 1 and 4.
00:24
We're told that there's a mass density, row, of ky squared.
00:32
Thus, given that mass density in that area, we can figure out what these values are.
00:38
The mass is the integral over the region of the mass density, whereas the x center of mass is the integral over x times the mass density over the region, and the center of mass in the y coordinate is y times the density integrated over the region.
00:59
Let's start with the mass of the object.
01:04
This should be the area integral over the region, d, of that mass density, ky squared.
01:14
Since the region is a square with x values of 1 to 7 and y values 1 to 4, these represent the bounds of integration.
01:26
Integrating first with respect to x we end up with k y squared x from evaluated from one to seven which is equal to the integral of from one to four of k y squared six d y integrating here we have the integral of k y cubed over three integrated from one to four, which is equal to k y four to the third power, minus one to the third power, all over three, which should be equal to 63 over 3, which should be equal to 63 over 3, which should be equal to 20 ,000...