Evaluating ∭ xyz dV over Box [0,1]×[0,2]×[0,3]

Calculus 3 · Axiom Academy

Evaluate a triple integral over a rectangular box by splitting it into a product of three single integrals. Evaluate the triple integral , where E is the rectangular box — that is, , , and . The region is a box, so every bound is a constant. Nice work — you evaluated a triple integral over a box by peeling it off one variable at a time. Constant bounds: over a box every limit is a number, so the order of integration never changes the answer. Work inside-out: integrate the innermost variable first, treating the others as constants, then move outward. It separates: since xyz factors and the bounds are constant, — a fast way to check the work. Master this on boxes first; the same inside-out method handles regions where the bounds depend on the other variables.

This is the written version of the interactive lesson above. See the full Calculus 3 course.