These are supposed to be triple integrals.
You should switch to spherical co-ordinates with the region being
, and .
I must find the z centroid of a hemisphere with radius a. It's base is on the x-y plane and its dome extends up the z axis. I am using the following equations to determine the centroid.
I am using and and integrating from 0 to a
I know the answer is supposed to be Where did I mess up?
The hemisphere is center on the origin so I know that the x and y centroid are 0. Is this what you are refering to by triple integrals?
I also would like to be able to do this in cartesian if that is possible as that was that coordinates the problem specified.
If I write a in terms of z I get
Then when ever I try to get the integrand in terms of one variable I wind back up at
For example the bound for z would be . Then the (x,y) base is a circle of radius a, also screaming out for trig substitution.
The other bounds of integration would be and . Which is begging for polar, i.e., trig substitution.