Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region.
-F(x, y, z) = 9x + 8y + 10z over the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1
A) $\frac { 3 } { 2 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 9 } { 8 }$
D) 1

Question

Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region.
-F(x, y, z) = 9x + 8y + 10z over the tetrahedron bounded by the coordinate planes and the plane x + y + z = 1 
A) $\frac { 3 } { 2 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 9 } { 8 }$
D) 1

Quizplus · Accepted Answer

Using a CAS integration utility, we can evaluate the integral as follows:

∫∫∫ 9x + 8y + 10z dV

where the limits of integration are:
0 ≤ z ≤ 1-x-y
0 ≤ y ≤ 1-x
0 ≤ x ≤ 1

Substituting the limits of integration, we get:

∫0^1 ∫0^(1-x) ∫0^(1-x-y) (9x + 8y + 10z) dz dy dx

After integrating with respect to z, we get:

∫0^1 ∫0^(1-x) (9x + 8y + 10(1-x-y))(1-x-y) dy dx

After integrating with respect to y, we get:

∫0^1 (9x + 8(1-x)/2 + 10(1-x)/2)(1-x)/2 dx

Simplifying, we get:

∫0^1 (9x + 9)x/4 dx

Evaluating this integral, we get:

(81/8)/2 = 9/8

Therefore, the correct answer is C) $\frac { 9 } { 8 }$.

Use a CAS Integration Utility to Evaluate the Triple Integral 32\frac { 3 } { 2 }23​

Use a CAS Integration Utility to Evaluate the Triple Integral $\frac { 3 } { 2 }$