Exercise 1077Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

Mass of a cube with variable density

#triple integral#density#cube

Statement

Compute the mass of $D=[0,1]^3$ with density $\mu(x,y,z)=x^2y+xz^2$ .

Hint

Integrate the density over the cube by separable terms.

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Setup.
$M=\int_0^1\int_0^1\int_0^1(x^2y+xz^2)\,dz\,dy\,dx.$

Inner in $z$ . $x^2y+\frac x3$ .

In $y$ . $\frac{x^2}{2}+\frac x3$ .

In $x$ . $M=\frac16+\frac16=\frac13$ .

Conclusion. $M=1/3$ .