Approximating Integrals

In calculus, we often encounter functions that are difficult or impossible to integrate using standard techniques. While we have a robust set of tools for finding derivatives of complex functions, integration sometimes requires us to use approximation methods. One such method is the Riemann Sum, which allows us to approximate the area under a curve by summing up the areas of multiple rectangles.

For example, consider the function [latex]g(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}[/latex]. While it is simple in form, it does not have an antiderivative expressible in terms of elementary functions. To approximate the definite integral [latex]\int_0^1 g(x) \, dx[/latex], we can use a Riemann Sum. By dividing the interval [latex][0,1][/latex] into subintervals and calculating the sum of the areas of the rectangles, we can find an approximate value for the integral.

We can further refine our approximation by calculating both left-endpoint and right-endpoint Riemann Sums. The left-endpoint sum tends to overestimate the integral, while the right-endpoint sum underestimates it. By combining these estimates, we can provide upper and lower bounds for the true value of the integral, offering a clearer picture of its range. This method of approximation is especially useful when dealing with integrals of functions without elementary antiderivatives, ensuring we can still make accurate and meaningful calculations.