Let f : R --> R be an odd function. Explain why, for any positive number a, the integral from -a to a of f(x) = 0.

To explain why the integral of an odd function from -a to a is equal to 0, we need to understand what it means for a function to be odd and how it affects the integration process.

A function f(x) is said to be odd if it satisfies the property f(-x) = -f(x) for all values of x in its domain. In other words, if you replace x with its negative counterpart and negate the output, you get the same function back.

Now, let's consider the integral from -a to a of an odd function f(x). The integral represents the signed area between the graph of the function and the x-axis over the interval from -a to a.

When we integrate an odd function over equal and opposite intervals such as -a to a, we are essentially summing up positive and negative areas that cancel each other out due to the symmetry of the function.

Since f(x) is an odd function, the value of the function at x and -x is related by f(-x) = -f(x). As a result, when you consider the signed area under the curve from -a to a, each positive area above the x-axis is matched by an equal negative area below the x-axis.

This cancellation of positive and negative areas leads to the conclusion that the total signed area between f(x) and the x-axis over the interval from -a to a is equal to zero.

Therefore, for any positive number a, the integral from -a to a of an odd function f(x) is equal to 0.