00:01
In this problem, we're given a region bounded by y equals x squared and y equals x plus 12.
00:06
Our job is to find the centroid of that region.
00:08
So we start by finding the area of the region.
00:12
But first we have to sketch the region.
00:14
We're going to find out the intersections of the parabola and the line.
00:17
We have done that.
00:19
The intersections are the points negative 3, 9, and 416.
00:23
So we have r, and we can now find its area.
00:26
We'll take the top boundary and subtract the bottom boundary and integrate.
00:31
Here's what we've done.
00:32
We're going to integrate from negative 3 to 4, top boundary minus bottom boundary.
00:37
There's the antiderivative, and we substitute, and see that the exact area of r is 57 and 1 -6th, which we've written in decimal here.
00:46
We need that area to find both the x -cordinent and the y -cordant of the central way.
00:51
We've set up the first part of the x -cordant, x -bar.
00:54
We have to divide by the area, and we're going to integrate now the function x times the difference and those boundaries.
01:01
So we have distributed the x...