In the first problem you are given and are asked to find it flux out of the ball . (you have but that is corrected in the integral so I assume that was a typo).
You do the integral by writing it as and then try to calculate , , and . You realize, do you not that those are just the coordinates of the center of the figure? Here that figure is a sphere with center at (2, 0, 3) so you should be able to write down, with no calculation at all, that , , and .
I have no idea at all how you got " "! You understand, don't you, that the x coordinate of the centroid of a figure is a number not a function of x? And it certainly is NOT the case that !
I give sincere thanks to HallsofIvy who has generously presented me with the most promising answer imaginable. I address you directly, HallsofIvy:
I do have received teaching in pre-calculus math: I was just insecure about the term "centroid".
Student, Chemistry (Bachelor) at University of Southern Denmark (SDU).
So the flux out of the ball is the following: