Double intergration over rectangular coordinates

[arun_gm]

Consider a 3d rectangular coordinate system.
A semicircle of radius a is drawn on the x-axis with its centre at the origin. An identical circle is drawn upon the y-axis. The two circles are oriented perpendicular to one another.
If we use Riemann's sums, multiplying the two curves with one another should give us a 3D hemisphere on the x-y plane.
The areas under each 2D curve is [MATH]\pi\frac{a^2}{2}[/MATH]The volume of the hemisphere should be the product of the areas under the two curves, which it obviously isn't.
Where am I going wrong?

 

[HallsofIvy]

I don't know what you mean by "multiplying the two curves with one another".
How do you "multiply" curves?

 

[Dr.Peterson]

Consider a 3d rectangular coordinate system.
A semicircle of radius a is drawn on the x-axis with its centre at the origin. An identical circle is drawn upon the y-axis. The two circles are oriented perpendicular to one another.
If we use Riemann's sums, multiplying the two curves with one another should give us a 3D hemisphere on the x-y plane.
The areas under each 2D curve is [MATH]\pi\frac{a^2}{2}[/MATH]The volume of the hemisphere should be the product of the areas under the two curves, which it obviously isn't.
Where am I going wrong?

I don't know how to draw a semicircle on the x-axis. It has to be on a plane, not on a line! What plane is each semicircle in? And where are they?

At the least, you should show us a picture of what you have in mind. Then show the Riemann sums you are referring to, so we can understand what you mean. Also, why should the volume be the product of two areas? The units aren't even right. I suspect your error is in starting from false assumptions.