triple integral

$\iiint_E f\,dV$ — the limit of a Riemann sum over small boxes filling a solid, one dimension up from a double integral.

A triple integral $\iiint_E f(x,y,z)\,dV$ sums a function over a solid $E$ by chopping it into small boxes of volume $dV$ , multiplying each by $f$ evaluated there, and summing as the boxes shrink to zero volume. Over a rectangular box, that sum peels apart into three ordinary single-variable integrals computed from the inside out, exactly like a double integral: the innermost integral treats the other two variables as constants, and each integration strips away one variable until a single number remains.

Examples

$\iiint_E (x+2z)\,dV$ over $0\le x\le1$ , $0\le y\le1$ , $0\le z\le1-x$ evaluates to $\tfrac12$ .