Calculating Average Height of a Constricted Hemisphere

[jimbo71]

Homework Statement

find the anverage heigh of z=sqrt(a^2-x^2-y^2) constricted by the cone x^2+y^2<=a^2
in the xy plane

Homework Equations

Average Height =(1/area)*double integral of region of [z]drdpheta

The Attempt at a Solution

I really have no idea how to solve this problem can you please point me in the right direction

 

[quantumdude]

How about converting your Cartesian functions over to cylindrical coordinates?

 

[jimbo71]

converted the cartesian equations to polar and used the 1/area*double integral of region R [z]rdrdpheta. I am having much difficulty integrating r*sqrt(a^2-r^2). i tried a trig substitution but don't know how to finish from there. please help me!

 

[cartonn30gel]

replace z in that "1/area*double integral of region R [z]rdrdpheta" you wrote by what it is equal to looking at the surface. then you can use polar coordinates.

 

[Dick]

If it's any help, you aren't integrating over a cone, you are integrating over a hemisphere.