Approximate Integration

Approximate integration is used when the exact value of a definite integral cannot be calculated.

This typically arises in two main cases: when the function's antiderivative is unknown or does not exist in a closed form, and when the function is composed of empirical data, such as a set of discrete points from an experiment, rather than a continuous formula.

In such situations, definite integrals are estimated using Riemann sums, which divide the interval into n equal subintervals of width Δx.

For each subinterval, a rectangle is constructed, with its height set by the function’s value at a specific point within that subinterval.

In the left endpoint approximation L_n, each rectangle’s height is set by the function’s value at the left end.

If the function is increasing, this method underestimates the area; if the function is decreasing, it overestimates.

The right endpoint approximation R_n, uses the right end of each subinterval. This method overestimates the area if the function is increasing and vice versa.

These methods use basic geometry—adding rectangle areas—to estimate integrals of complex or unknown functions.