Question

how do i find the centroid of the region in the first quadrant bounded by the given curves.
y = x^2, x = y^2

Answer 1

I can't tell you how to do this using calculus. But I can sketch the analysis required for getting a numerical solution. Please see the attached diagram that I made using geogrebra.

First of all, the two curves are inverses of one another; therefore, their graphs are symmetric around the line y=x, as I'm sure you're aware. This implies that the centroid lies on the line y=x and that, if we could rotate this space through -45 degrees, we need the point where half the mass of the region is to the left of the point. By symmetry we need no consider both curves for this, we need consider only the one that is most convenient for calculation (not that either of them is all that convenient as far as I can tell).

We need to think of a variable of integration going from 0 to sqrt(2), ie, over the length of the line y=x enclosed by the region of interest. A typical point A on the line is (z/sqrt(2)), z being the variable of integration, 0 le z le sqrt (2). A little gruesome algebra gives us the typical point B through A and perpendicular to y=x. Now we can calculate the distance between A and B which is the function needing integration.

We integrate this function over the entire range of z to obtain the area of half the region, then use some form of optimisation to find the value of z that gives half of this area. This is the value of z that gives the centroid as (z/sqrt(2),z/sqrt(2)).