Step 1: Understanding the Concept:
This is a definite integral with symmetric limits of integration, from \( -a \) to \( a \) (where \( a = 3 \)).
In such cases, we should immediately check the parity of the function (whether it is even or odd).
An odd function \( f(x) \) satisfies \( f(-x) = -f(x) \).
An even function \( f(x) \) satisfies \( f(-x) = f(x) \).
Step 2: Key Formula or Approach:
The properties of definite integrals for symmetric limits are:
1. If \( f(x) \) is an odd function, then \( \int_{-a}^{a} f(x) dx = 0 \).
2. If \( f(x) \) is an even function, then \( \int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx \).
Step 3: Detailed Explanation:
Let the integrand be \( f(x) = x^{3} - x \).
Check the parity by replacing \( x \) with \( -x \):
\[ f(-x) = (-x)^{3} - (-x) \]
Using the fact that \( (-x)^{n} = -x^{n} \) for odd \( n \):
\[ f(-x) = -x^{3} + x \]
Factor out a negative sign:
\[ f(-x) = -(x^{3} - x) \]
\[ f(-x) = -f(x) \]
Since \( f(-x) = -f(x) \), the function \( f(x) \) is strictly an odd function.
Applying the property for odd functions with symmetric limits:
\[ \int_{-3}^{3} (x^{3} - x) dx = 0 \]
Verification by standard integration:
\[ I = \left[ \frac{x^{4}}{4} - \frac{x^{2}}{2} \right]_{-3}^{3} \]
\[ I = \left( \frac{3^{4}}{4} - \frac{3^{2}}{2} \right) - \left( \frac{(-3)^{4}}{4} - \frac{(-3)^{2}}{2} \right) \]
\[ I = \left( \frac{81}{4} - \frac{9}{2} \right) - \left( \frac{81}{4} - \frac{9}{2} \right) \]
\[ I = 0 \]
The geometric interpretation is that the area above the x-axis on one side exactly cancels out the area below the x-axis on the other side due to origin symmetry.
Step 4: Final Answer:
The final value of the integral is 0.