Exercise · ZW-TRIPLE-INTEGRAL-004 · ★☆☆ Basic
The tetrahedron \(E\) is bounded by the coordinate planes and \(x+y+z=1\). Its volume \(\int_0^1\!\int_0^{1-x}\!\int_0^{1-x-y}dz\,dy\,dx\) is:
Choose one option.
Show answer
- B. \(\tfrac16\)
Explanation
The inner integral gives \(1-x-y\); then \(\int_0^{1-x}(1-x-y)\,dy=\tfrac{(1-x)^2}{2}\) and \(\int_0^1\tfrac{(1-x)^2}{2}dx=\tfrac16\). It matches \(\tfrac13\cdot\tfrac12\cdot1\) (base times height over three).
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-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: