The Volume Riemann Integral
\[f(x,y,z)\]
defined on a region \[V \subset \mathbb{R}^3\]
.We can divide
\[V\]
into \[N\]
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
\[f(x,y,z)\]
is continuous throughout \[V\]
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
\[f(x,y,z)=x^3\]
then \[f\]
is continuous on the unit cube \[0 \leq xyz \leq 1\]
and the Riemann Sum is \[\int^1_0 \int^1_0 \int^1_0 x^3 dxdydy = \frac{1}{4}\]
