The Volume Riemann Integral

Suppose we have a function  
  defined on a region  
\[V \subset \mathbb{R}^3\]
We can divide  
  small volumes  
\[\delta V_1 , ... \ \delta V_N\]
Then if  
\[(x_K , y_K , z_K) \in V_K \]
  we can define  
\[F= \sum_K f(x_K , y_K , z_K ) \delta V_K \]

Suppose that  
  is continuous throughout  
  and on its bounday, then  
\[lim_{MAX(\delta V_K) \rightarrow 0} \sum_K f(x_K ,y_K , z_K ) \delta_K \]
  exists and is equal to  
\[\int_V f(x,y,z) dV\]

This is called the Volum Riemann Sum.
Example: If  
  is continuous on the unit cube  
\[0 \leq xyz \leq 1\]
  and the Riemann Sum is  
\[\int^1_0 \int^1_0 \int^1_0 x^3 dxdydy = \frac{1}{4}\]