Question:
Find the average z coordinate of all the points on AND within a hemisphere of radius 4 centered at the origin, and with it's base in the xy-plane.
So I am assuming the function will be x^2 + y^2 = 4
The bounds in the y axis from 0 to 4 while in the x the bound is the function (correct??)
I am thinking this is just a tricky way to ask me to setup a double integral but I am confused on how to go about doing this.
Thank you for the help.
The important part is the points within the hemisphere. The volume of the surface of the hemisphere is zero, so it makes no contribution to the overall average.
The problem you appear to be asking is equivalent to finding the distance to the center of mass of a uniformly dense hemisphere, measured from the circular base of the hemisphere.
I would integrate it by the "disk" method, each disk parallel to the $x,y$ plane. All points of a disk have the same value of $z$, so the integral looks like this: $$\int_0^R z \cdot A(z)\, dz$$ where $R$ is the radius of the hemisphere (in this case, $R=4$) and $A(z)$ is the area of the disk within the sphere at distance $z$ from the $x,y$ plane.
I suggest this answer for further details.
