Question

Evaluate ∫∫∫ 12y-8x dV  in the region behind y=10-2z and bounded by z=2x, z=5 and x=0.

(a) 1

(b) \(\frac{35}{63}\)

(c) \(\frac{3125}{16}\)

(d) \(\frac{3125}{6}\)

I have been asked this question during an interview.

My question is based upon Applications of Triple Integral in chapter Multiple Integrals of Engineering Mathematics

Answer 1

Correct choice is (d) \(\frac{3125}{6}\)

Explanation: We know

From the boundary conditions,

0 < y < 10-2z

0 < x < \(\frac{z}{2}\)

0 < z < 5

Applying these limits on the Triple Integral as follows

\(\int\int\int_{0}^{10-2z} 12y-8x dy dx dz\)

\(=\int\int_{0}^{\frac{\pi}{2}}6 (10-2z)^2-8x(10-2z) dx dz\)

\(=∫_{0}^{5} 14z^3-130z^2+300z dz\)

\(=\frac{3125}{6}\)

Thus the answer is \(=\frac{3125}{6}\).