If this is the upper hemisphere, you would do better to integrate in the order "dzdydx" rather than "dxdydz". Using Cartesian coordiates, and itegrating in the order "dzdydx", the limits on the outer integral have to cover all possible x-value so run from -a to a. To determine the y limits, observe that the hemisphere projects down to the xy-axis as the region inside or so y runs from to . Finally, since we have the upper hemisphere of , which is equivalent to , z will go from 0 to .
That is, the integral would be .
If you want to use "spherical coorinates" then, at least, you need to understand what the coordinates mean! For any given point, (x, y, z), (not r) is the staright line distance from (0, 0, 0) to (x, y, z), is the angle, in the xy- plane, between the line from (0, 0) to (x, y) and the x-axis, so from 0 to to cover the entire plane, and is the angle the line from (0, 0, 0) to (x, y, z) makes with the positive z-axis, so from 0 to to cover all the way from the positive z-axis to the negative z-axis.
So for the upper hemi-sphere, with center at (0, 0, 0) and radius a, goes from 0 to a, goes from 0 to , and goes from 0 to .