Triple integral over the unit cube
∫₀¹∫₀¹∫₀¹ xyz dz dy dx = 1/8 ≈ 0.125. A worked triple integral in Cartesian coordinates, with an editable calculator preloaded with the limits.
∫₀¹∫₀¹∫₀¹ xyz dz dy dx = 1/8 ≈ 0.125. The limits are all constants, so the integral separates into three identical single integrals of ½ each.
dV = dz dy dx
In terms of x, y, z
Limits of integration
Inner limits may use the outer variables — e.g. 1-x, sqrt(1-x^2)
The setup
∫₀¹∫₀¹∫₀¹ xyz dz dy dx = 1/8 ≈ 0.125
The limits are all constants, so the integral separates into three identical single integrals of ½ each.
Coordinate system: Cartesian. The integrand is x*y*z, integrated from 0 to 1 innermost, then 0 to 1, then 0 to 1.
More worked triple integrals
Frequently asked questions
What is the value of ∫₀¹∫₀¹∫₀¹ xyz dz dy dx? ▾
It evaluates to 1/8 ≈ 0.125. The limits are all constants, so the integral separates into three identical single integrals of ½ each.
Why Cartesian coordinates for this one? ▾
The region is bounded by planes and graphs, so rectangular limits describe it directly with no change of variables needed.
Do I need to include the Jacobian myself? ▾
There is no Jacobian in Cartesian coordinates — dV is simply dz dy dx.
About this Calculator
∫₀¹∫₀¹∫₀¹ xyz dz dy dx = 1/8 ≈ 0.125. A worked triple integral in Cartesian coordinates, with an editable calculator preloaded with the limits.