What is the Limit of the Centroid of an Area Bounded by the x-Axis and 1-x^n?

[tinylights]

Homework Statement

Okay, so the idea here is to take the centroid bounded by the x-axis and 1-x^n. N should be an even and positive integer. We should take the limit as n approaches infinity of both the x- and y-coords of the centroid, hopefully ending up with (0, 1/2).

Homework Equations

Formula for a centroid: A = integral of f(x), Mx = integral of xf(x), My = integral of f(x)^2 over 2. All of them should be definite integrals with the same "a" and "b" - I picked 0 and 1 and multiplied the whole thing by two since it appears to be an even function.

Then you divide Mx/A for your x-coord, My/A for your y-coord, and take the limit.

The Attempt at a Solution

See above. I am doing these steps over and over again but keep ending up with the same weird things. I get (n+1)/2(n+2) for my Mx/A, and this long convoluted thing for My/A - 1-2ⁿ⁺¹/n+1 + 1/2n+1 / 2(1 - 1/n+1). This would be fine except the limit for Mx/A seems to be 1/2, when it really should be 0, and I'm certainly not getting a limit of 1/2 for My/A.

Thanks for your help guys.

 

[MostlyHarmless]

Without seeing all your steps, its kind of hard to see where you could have gone wrong. It may be algebra. If you could; could you show how you got those expressions for Mx/A and My/A? Pictures of your work would suffice.