Just need a few short questions verified:
In finding the centroid of the solid region of uniform density 'd' between the xy plane and the paraboloid z = 4 - (x^2 + y^2), the paraboloid opens down and we have a cicle about the xy plane, thus we can use cylindrical coordinates right? The big question is does this paraboloid only have symmetry with respect to all the three coordinate planes passing through the z axis? Therefore when finding the centroid (x,y,z) x=y=0 due to symmetry and we only need to find the moment with respect to z. Right?
This is an improper integral question.
I = double integral [ln(x)ln(y)/((x^2)*(y^2))] dA,
D = [1, infinity] x [1, infinity], is it convergent or divergent?
My question is can we simplify the integral by splitting it up with respect to x and y, that is, can we integrate ln(x)/x^2 with respect to x and multiply it by the integral ln(y)/y^2 with respect to y? This would be a lot simpler!
The other question is: I know that ln(b) = + infinity, as b --> + infinity.
Does (ln(b))/b approach zero as b --> + infinity?
My confusion is if ln(b) = infinity, then we have for ln(b)/b = infinity/infinity = 1. However, when I plug actual numbers in for infinity, say b = 1000000, I get: ln(1000000)/1000000 = 13.81/1000000 = 1.38x10^-5 = 0. If this is try, I get a convergent improper integral with a value of 1.