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?
Thanks
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
Thanks again
You can do this via (x,y,z) but to solve the integrals you will need to various trig substitutions. It's smarter to switch to spherical immediately.
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.
Here's the problem worked in a bit more detail, hope it reads:
Calculus: Centroid of a Hemisphere, Math 251