## The Volume Riemann Integral

Suppose we have a function
$f(x,y,z)$
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
$\int^1_0 \int^1_0 \int^1_0 x^3 dxdydy = \frac{1}{4}$