Question

The value of $\iiint_B (x + 2y + 3z)^2 \, dx \, dy \, dz$ is, where $B$ is the box $[0, 1] \times \left[-\frac{1}{2}, 0\right] \times \left[0, \frac{1}{3}\right]$

• A. 0
• B. $\frac{1}{12}$
• C. $\frac{1}{3}$
• D. 1

Hint:

For linear combination integrands $f(a x + b y + c z)$, each integration step introduces a scaling factor of $1/a, 1/b, 1/c$ from the chain rule. Here $1 \cdot 2 \cdot 3 = 6$, and dividing during integration reduces the power integral cleanly!

Answer 1

The Correct Option is B