Symmetry, parity and periodicity

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

On a symmetric interval an integrable odd function has integral zero; an even function lets you double the positive half. Periodicity splits intervals into full periods and remainders.

Step-by-step example

1. Check that x³ is odd.

   $$f(-x)=-f(x)$$

2. The interval is symmetric and the function is continuous.

   $$I=\int_{-2}^2x^3\,dx$$

3. The contributions cancel.

   $$I=0$$