Mass of a solid with density ρ = z
∫₀²∫₀²∫₀² z dz dy dx = 8 ≈ 8. A worked triple integral in Cartesian coordinates, with an editable calculator preloaded with the limits.
∫₀²∫₀²∫₀² z dz dy dx = 8 ≈ 8. With density increasing upward, the mass is the density integrated over the volume.
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
∫₀²∫₀²∫₀² z dz dy dx = 8 ≈ 8
With density increasing upward, the mass is the density integrated over the volume.
Coordinate system: Cartesian. The integrand is z, integrated from 0 to 2 innermost, then 0 to 2, then 0 to 2.
More worked triple integrals
Frequently asked questions
What is the value of ∫₀²∫₀²∫₀² z dz dy dx? ▾
It evaluates to 8 ≈ 8. With density increasing upward, the mass is the density integrated over the volume.
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
∫₀²∫₀²∫₀² z dz dy dx = 8 ≈ 8. A worked triple integral in Cartesian coordinates, with an editable calculator preloaded with the limits.