Topic: Engineering Mathematics
The value of the triple integral $\iint_{1}^{2}\int_{0}^{3}\int_{-3}^{3}2xy-x^3 dx dy dz$
- A. 9
- B. 18
- C. 0
- D. None of the above
Answer: C. 0
Explanation: Assuming the integrand is $2xy - x^3$ (based on symmetrical bounds logic common to such problems), the inner integral evaluates with respect to $x$ from $-3$ to $3$.
The function $f(x) = 2xy - x^3$ is an odd function with respect to $x$ because $f(-x) = 2(-x)y - (-x)^3 = - (2xy - x^3) = -f(x)$.
The integral of any odd function over a symmetric interval $[-a, a]$ is $0$.
Therefore, $\int_{-3}^{3} (2xy - x^3) dx = 0$. Since the innermost integral evaluates to 0, the entire triple integral evaluates to 0.