Find the centroid of region bounded by y = 8 - x^2, y = 0

[MarkSA]

Hello,

1) Find the centroid of the region bounded by the curves:
y = 8 - x^2, y = 0

I'm confused about how to find the coordinates of the centroid. I graphed this and it appears to be a parbola opening downward, vertexs at (0,8) and crosses the x-axis at points -sqrt(8) and +sqrt(8). It looks symmetric about x = 0...

I'm thinking that the x coordinate of the centroid would be 0 since it's symmetric. But that doesn't appear to be the case when I try to integrate using the formula for x-bar. x-bar = M(sub y)/m
M(sub y) = 32
m = 64*sqrt(2)/3
So I get x-bar = 3*sqrt(2)/4

Any idea where i'm going wrong at?
Thanks

 

[MarkSA]

Re: Find the centroid

Nevermind, I think I realized the problem.

I was trying to multiply the M(sub y) integral by 2 for symmetry, except since the formula for M(sub y) adds an x, that function is no longer symmetric...

If I don't try the symmetry trick then I get the proper answe for x-bar of 0.

For y-bar I get 16/5 as an answer.

So the centroid is at (0,16/5) if I did everything correctly...

 

[wjm11]

Re: Find the centroid

the proper answe for x-bar of 0.

For y-bar I get 16/5 as an answer.

So the centroid is at (0,16/5) if I did everything correctly...

You are correct.