Discussion
On the symmetric interval from \( -a \) to \( a \), the cube \( x^{3} \) integrates to
Answer: C
The integrand is an odd function, \( (-x)^{3} = -x^{3} \), and the interval is symmetric about the origin, so the two halves cancel:
\[ \int_{-a}^{a} x^{3}dx \;=\; \left[\frac{x^{4}}{4}\right]_{-a}^{a} \;=\; \frac{a^{4}}{4} - \frac{a^{4}}{4} \;=\; 0. \]
For an even integrand the halves would add instead, giving twice the integral over half the range
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