∫∫∫ (x² + y² + z²) over the unit cube
∫₀¹∫₀¹∫₀¹ (x² + y² + z²) dz dy dx = 1 ≈ 1. A worked triple integral in Cartesian coordinates, with an editable calculator preloaded with the limits.
∫₀¹∫₀¹∫₀¹ (x² + y² + z²) dz dy dx = 1 ≈ 1. Three identical contributions of 1/3 give exactly 1 — a quick mass-moment style calculation.
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 (x² + y² + z²) dz dy dx = 1 ≈ 1
Three identical contributions of 1/3 give exactly 1 — a quick mass-moment style calculation.
Coordinate system: Cartesian. The integrand is x^2+y^2+z^2, 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 (x² + y² + z²) dz dy dx? ▾
It evaluates to 1 ≈ 1. Three identical contributions of 1/3 give exactly 1 — a quick mass-moment style calculation.
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 (x² + y² + z²) dz dy dx = 1 ≈ 1. A worked triple integral in Cartesian coordinates, with an editable calculator preloaded with the limits.