Exercise · ZW-TRIPLE-INTEGRAL-009 · ★★★ Advanced
The average value \(\tfrac1{V(E)}\iiint_E f\,dV\) of \(f=x+y+z\) over the unit cube \(E=[0,1]^3\) is:
Choose one option.
Show answer
- C. \(\tfrac32\)
Explanation
The volume is \(1\) and the integral is \(\tfrac12+\tfrac12+\tfrac12=\tfrac32\), so the average is \(\tfrac32\). The value \(3\) would be the maximum of \(f\), not its average.
Review the concept: Triple integral
The triple integral of a function \(f\) over a solid \(E\) of space is the limit of the sums of the values of \(f\) times the volume \(\Delta V\) of ever finer cells covering \(E\).
\[\iiint_E f\,dV=\lim_{\max\Delta V_i\to0}\sum_i f(x_i^*,y_i^*,z_i^*)\,\Delta V_i\]
More practice problems: triple integral
- ZW-TRIPLE-INTEGRAL-001 Compute iiint_B 1 dV over the box B=[0,1]×[0,2]×[0,3].
- ZW-TRIPLE-INTEGRAL-002 Compute ∫_0^1\!∫_0^1\!∫_0^1 xyz dz dy dx.
- ZW-TRIPLE-INTEGRAL-003 Compute ∫_0^1\!∫_0^2\!∫_0^3 (x+y+z) dz dy dx.
- ZW-TRIPLE-INTEGRAL-004 The tetrahedron E is bounded by the coordinate planes and x+y+z=1. Its volume ∫_0^1\!∫_0^1-x\!∫_0^1…
- ZW-TRIPLE-INTEGRAL-005 With the same tetrahedron E, compute iiint_E z dV.
- ZW-TRIPLE-INTEGRAL-006 A cube [0,2]^3 has density rho(x,y,z)=x. Its mass iiintrho dV is:
- ZW-TRIPLE-INTEGRAL-007 Compute ∫_0^1\!∫_0^x\!∫_0^y x dz dy dx.
- ZW-TRIPLE-INTEGRAL-008 Find the volume of the solid in the first octant under the plane x+2y+3z=6.