Triple Integrals

_V f dV — 3D iterated integration, with cylindrical (r) and spherical ( ^2 ) Jacobians.

The idea

A triple integral adds up a function of three variables over a solid region of space. Take a solid $V$, chop it into small boxes of volume $\Delta V$, weight each box by the value of $f$ somewhere inside it, and add: $\iiint_V f(x, y, z)\, dV = \lim_{\Delta V \to 0} \sum f(x_i, y_i, z_i)\, \Delta V.$ The construction is the double integral's, with boxes in place of patches of area; nothing in it depends on the dimension.

We evaluate a triple integral one variable at a time, as three ordinary single integrals nested inside each other. On a rectangular box all six limits are constants, and the integrations may be performed in any order.

On any other solid the limits are not constants. Fix the outermost variable, and the solid's cross-section at that value is a two-dimensional region; fix the next, and what remains is a segment. So the innermost limits may depend on both variables outside them, and the middle limits on the outermost. Setting those limits up correctly, not the integration itself, is where the work of a triple integral usually lies.

Ways to work on it

• Walkthrough. From boxes to the unit tetrahedron — set up limits, compute volume.
• Practice. Split a triple integral over a box into a product of single integrals.
• Hardest. Change to cylindrical or spherical coordinates and compute a volume.

Not sure where to start? Take the ten-question placement test.