24. Triple Integrals

Consider a 3D solid with a variable density given by $\large \rho(x,y,z)$. How do we find the mass of the solid? Let's look at one little box of the solid as pictured:

^{^{Image from: Briggs Calculus, Third Edition}}

Then we can say that the mass of that little piece is approximately $\large \rho(x_i,y_j,z_k)\Delta x\Delta y\Delta z$ (the density at the point times the volume of the small box). By now, we know what's coming...we add up all the boxes, get a Riemann sum, and take the limit to get what we call a triple integral:

$$\large \text{Total Mass }=\iiint_V\rho(x,y,z)\, dV=\lim_{L,M,N\to\infty}\sum_{i=1}^L\sum_{j=1}^M\sum_{k=1}^N\rho (x_i,y_j,z_k)\Delta x\Delta y\Delta z$$

Note: $\large \iiint_D f(x,y,z)dV$ is not an area or a volume or any geometric quantity. Our best intuition is to think of the integrand $\large f$ as a density. Then the triple integral represents a total amount.

Simplest Case: D is a Box

If the region of integration $\large D$ is a box $\large [a,b]\times[c,d]\times[e,g]$, then the triple integral of $\large f$ over $\large D$ is an iterated integral:

$$\large \iiint_D f(x,y,z)\, dV = \int_a^b\int_c^d\int_e^g f(x,y,z) \, dzdydx$$

This can be done in any order.

More Interesting Regions

A more interesting region may look like this:

zfirst

^{^{Image from: https://math.libretexts.org/Bookshelves/Calculus/Calculus_%28OpenStax%29/15%3A_Multiple_Integration/15.04%3A_Triple_Integrals}}

We will call this type of region a z-first region. We can integrate first in $\large z$, and then we will be left with a double integral:

$$\large \iiint_V f(x,y,z) \, dV = \iint_R\left(\int_{z=u_1(x,y)}^{z=u_2(x,y)}f(x,y,z)\, dz\right)\, dA$$

We can similarly define $\large x$-first and $\large y$-first regions, though we will stick to $\large z$-first in this class. Here is an example of a triple integral problem:

The steps for triple integral problems are:

Try to visualize $\large D$, the region of integration. This is often challenging!
Integrate in $\large z$.
We now have a double integral in $\large x$ and $\large y$ over the region $\large R$, which is the shadow of $\large D$ down onto the $\large xy$-plane. Sketch $\large R$. This may involve finding the intersection of the surfaces that define $\large D$.
Decide how to approach the double integral ($\large x$-first? $\large y$-first? Polar coordinates?)
Do the double integral