Exercise · ZW-TRIPLE-INTEGRAL-001 · ★☆☆ Basic
Compute \(\iiint_B 1\,dV\) over the box \(B=[0,1]\times[0,2]\times[0,3]\).
Choose one option.
Show answer
- C. \(6\)
Explanation
With \(f=1\) the triple integral is the volume of the box: \(1\cdot2\cdot3=6\).
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-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.
- ZW-TRIPLE-INTEGRAL-009 The average value tfrac1V(E)iiint_E f dV of f=x+y+z over the unit cube E=[0,1]^3 is: