Change of Order of Integration
Text Applied to continuous functions of three variables, Fubini's Theorem says that there are six different ways to evaluate integrals.
For a function f(x, y, z) with domain R such that a ≤ x ≤ b, c ≤ y ≤ d, k ≤ z ≤ l,
∫_{a}^{b}∫_{c}^{d}∫_{k}^{l}f(x, y, z)dzdydx =
∫_{a}^{b}∫_{k}^{l}∫_{c}^{d}f(x, y, z)dydzdx =
∫_{c}^{d}∫_{a}^{b}∫_{k}^{l}f(x, y, z)dzdxdy =
∫_{c}^{d}∫_{k}^{l}∫_{a}^{b}f(x, y, z)dxdzdy =
∫_{k}^{l}∫_{a}^{b}∫_{c}^{d}f(x, y, z)dydxdz =
∫_{k}^{l}∫_{c}^{d}∫_{a}^{b}f(x, y, z)dxdydz =
∫∫∫_{R}f(x, y, z)dV
Demos
Change of Order of Integration
 
To see that integration for functions of three variables can be done in six different ways, start "xStep", "yStep", and "zStep" at their minimum values. Now use the [>>] buttons to change each of these three variables to its maximum value. You could start by choosing any of the three to set to its maximum, then choose one of the remaining two, and finish with the remaining one. This gives 3 * 2 * 1 = 6 choices in the order of integration.

Exercises In the 2variable lab on Change of Order of Integration, there is a demo called "Slab Approximations." What would be the equivalent of a slab for integrals over three variables?
