Triple Integrals over Boxes

So we have looked at integrals of single variable real-valued functions and integrals of two variable real-valued functions. We will now go further and look at how we can evaluate integrals of a function of three variables.

First suppose that $w = f(x, y, z)$ is a three variable real-valued function that is defined on the rectangular box $B = \{ (x, y, z) : a ≤ x ≤ b, c ≤ y ≤ d, r ≤ z ≤ s \}$.

We will take this box $B$ and divide it into sub-boxes of equal dimension. This can be done by taking the interval $[a, b]$ and equally dividing it into $l$ subintervals $[x_{i-1}, x_i]$ with width $\Delta x$, then taking the interval $[c, d]$ and equally dividing it into $m$ subintervals $[y_{j-1}, y_j]$ with width $\Delta y$, and lastly taking the interval $[r, s]$ and equally dividing it into $n$ subintervals $[z_{k-1}, z_k ]$ with width $\Delta z$. In total, we obtain $l \cdot m \cdot n$ many sub-boxes with equal dimension.

Now we will identify each box $B_{ijk}$ as:

(1)

\begin{align} B_{ijk} = [x_{i-1}, x_i ] \times [y_{j-1}, y_j] \times [z_{k-1}, z_k ] \end{align}

Furthermore, each of these sub-boxes have volume $\Delta V$ and:

(2)

\begin{align} \quad \Delta V = \Delta x \Delta y \Delta z \end{align}

We are now ready to construct a triple Riemann sum and define the triple integral. Let $(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*)$ be a sample point contained in the box $B_{ijk}$. Thus the triple Riemann sum is:

(3)

\begin{align} \quad \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} f(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \Delta V \end{align}

Now if we take the limit as $l, m, n \to \infty$ of this triple Riemann sum, then we obtain the definition of the triple integral:

(4)

\begin{align} \quad \iiint_B f(x, y, z) \: dV = \lim_{l, m, n \to \infty} \sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \Delta V \end{align}

Definition: Let $w = f(x, y, z)$ be a three variable real-valued function defined on the box $B = \{ (x, y, z) : a ≤ x ≤ b, c ≤ y ≤ d, r ≤ z ≤ s \}$. Then the Triple Integral of $f$ over the box $B$ is defined as $\iiint_B f(x, y, z) \: dV = \lim_{l, m, n \to \infty} \sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \Delta V$ provided that this limit exists.

We will subsequently see on the Fubini's Theorem for Evaluating Triple Integrals over Rectangles that we can evaluate triple integrals over boxes in a similar manner for which we evaluated double integrals over rectangles.