Exercise · ZW-TRIPLE-INTEGRAL-008 · ★★☆ Intermediate
Find the volume of the solid in the first octant under the plane \(x+2y+3z=6\).
Choose one option.
Show answer
- A. \(6\)
Explanation
The axis intercepts are \(6,3,2\); the volume is \(\int_0^6\!\int_0^{(6-x)/2}\!\int_0^{(6-x-2y)/3}dz\,dy\,dx=\tfrac16\cdot6\cdot3\cdot2=6\).
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-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-009 The average value tfrac1V(E)iiint_E f dV of f=x+y+z over the unit cube E=[0,1]^3 is: