Consider the 3d rectangular coordinate system.

I have a semicircle on the x axis with radius=1, drawn between x=-1 and x=+1. I know that the area under this curve is [tex]\pi[/tex]/2

I draw another semicircle drawn on the y axis with radius=1, between y=-1 and y=+1. The area under this curve too is [tex]\pi[/tex]/2

By one of the rules of multivariable calculus, multiplying these two areas should give me the volume under a hemisphere, but I know it does not.

Where am I going wrong?