What does a triple integral represent?

It can be difficult to visualize what a triple integral represents, which is why in this video we’ll be answering the question, “What am I finding when I evaluate a triple integral?”

In order to answer this question, we’ll compare the triple integral to a double integral, so that we understand exactly how to transition from double integrals into triple integrals. Every piece of the double integral, like the integral, the bounds or limits of integration, the function which is the integrand, and the differential (usually dydx) will all translate into a corresponding piece of the triple integral.

The interesting thing about the triple integral is that it can be used in two ways. In contrast, single integrals only find area under the curve and double integrals only find volume under the surface. But triple integrals can be used to 1) find volume, just like the double integral, and to 2) find mass, when the volume of the region we’re interested in has variable density.

In this way, triple integrals let us do more than we were able to do with double integrals. We’re able to add in the extra dimension of variable density inside the volume, and based on that variable density, find the mass of the volume, as opposed to only being able to find volume, which is what we were limited to in the double integral.

If we want to describe double and triple integrals with words, we can say that for the double integral, we’re integrating a multivariable function f(x,y) over the region R which is defined for x on the interval [a,b] and for y on the interval [c,d], using vertical slices of volume, in order to find the total volume under the surface f(x,y) but above the xy-plane.

In contrast, we can say that for the triple integral, we’re integrating a multivariable function for density f(x,y,z) for the volume B which is defined for x on the interval [a,b] and for y on the interval [c,d] and for z on the interval [r,s], by slicing the volume in three direction to get tiny pieces (or boxes) of volume, in order to find the total mass of the volume.

triple integrals can be used to 1) find volume, just like the double integral, and to 2) find mass, when the volume of the region we’re interested in has variable density.

In this video you'll learn:

0:15 // Recap of what the double integral represents
1:22 // The triple integral has two uses (volume and mass)
1:45 // How to use the triple integral to find volume
8:59 // Why the triple integral does more than the double integral
11:19 // How to use the triple integral to find mass, when the volume has variable density