00:01
In this problem, we're given a region are with four boundaries, y equals x squared, y equals zero, which is the x -axis, x equals 1, and x equals 2.
00:10
Our job is to find the centroid of this region.
00:14
We've graphed the region with this upper boundary, y equals x squared, the lower boundary, the x -axis, and the two left and right boundaries.
00:23
We're looking for the centroid c.
00:25
The x -cordinate will denote by x -bar and the y -cordinate will denote by y -bar.
00:29
We need to calculate three integrals to find the centroid.
00:33
The first of those is the area of the region.
00:36
So we will integrate from 1 to 2 with respect to x, and we will take the top boundary of r and subtract to the bottom boundary.
00:47
So we need to find the antiderivative of x squared.
00:52
So we have 1 3rd x cubed, and we will substitute our limits of integration, and we conclude that the area of the region is 7 thirds.
01:07
We need that answer to find both the x and y coordinates of the centraloid.
01:11
The x coordinate divides by that area, and we have to integrate from 1 to 2, x times the difference in those boundaries...