Cavalieri's Principle

First off, we define our coordinate system.  
The positive X-axis is coming out of your screen towards you, 
the positive Y-axis heading off to your right, 
and the Z-axis is pointing straight up.  
Let's suppose we have some kind of solid body, say a peanut of height h.  
We define A(x) to be it's cross-sectional area at a height 
x (Assuming the object is sitting on the XY-plane).
The very simply, the volume of the afore mentioned peanut is:
            /h 
    volume= | A(x) dx 
            /0

whose approximating Reimann sum is:
         n-1
         \
         /__ A(c{i})(x{i+1}-xi)
         i=0 
Note: Braces indicate a subscript