2.6 Moments and Centers of Mass [STARTS AT 24:33] INSTRUCTOR: We are bounded between two functions. Just remember those rules that we have. So our mass is going to be equal to the integral of-- we don't actually know the region yet-- of 6 minus x squared minus 3 minus 2x dx. So let's go and sketch this. I've got 6 minus 2x. So say this is 6. 6 minus x squared is that function, 3 minus 2x. So for my sketch here, I've got something that looks kind of like that. So I could use technology to figure out where these intersect. One of them is at 3, and the other's at negative 1. So this is from negative 1 to 3. I'll go and do some of the algebra here. This is going to be integral from negative 1 to 3 of negative x squared plus 2x plus 3 dx, which is negative x cubed over 3 plus x squared plus 3x evaluated from negative 1 to 3, which is 32/3. To find Mx, it's going to be the integral from negative 1 to 3 of first function squared. Upper function, this here, 6 minus x squared. So we have 1/2 6 minus x squared, squared, minus 3 minus 2x squared dx. Can't leave that off. That's going to turn out to be 416/15, which is roughly 27.73 repeating. My, it's going to be the integral from negative 1 to 3 of x f of x minus g of x. That minus 3 minus 2x dx, which will be 32/3. So let's just point this out. We've got our mass is 32/3. We've got our Mx and our My. So our center of mass is going to be-- I guess I better do that first. x bar will be in My divided by m, which is 1. Y bar will be Mx, so 416/15 divided by our mass, 32/3, which is 13/5. So our center of mass is 1 comma 13/5. [ENDS AT 28:24]