Triple Integral Calculator

How it is calculated

Because the integrand is the same constant everywhere, the three nested integrals reduce to the constant times the box dimensions: ∭ k dV = k · (x1 − x0) · (y1 − y0) · (z1 − z0). Each factor in brackets is the edge length of the box along one axis, and their product is the volume V. All seven fields are plain numbers; if x, y and z are lengths in metres, the volume is in cubic metres and the integral carries the unit of k times m³. Each upper bound must be strictly greater than its lower bound, otherwise there is no result. k may be negative or zero, which gives a negative or zero integral. Results are shown with four decimals. The tool does not integrate functions of x, y or z.

Common mistakes

• Entering the bounds in the wrong order, for example x0 = 2 and x1 = 0. The tool gives no result, because each upper limit must exceed its lower limit.
• Treating a negative lower bound as a length. The edge is x1 − x0, so from −1 to 3 it is 4, not 3 or 2.
• Typing a function such as x²y instead of a constant in k. Only a constant integrand works; a variable integrand needs real integration, not k times volume.
• Mixing units between axes, such as x in centimetres and z in metres. The volume then has no meaning and k times it is wrong.
• Forgetting that the answer carries the unit of k times volume; it is not a volume itself unless k = 1.