Approximating Area with Riemann Sums

Approximate the area under the curve of the function $f(x) = 1 + x^2$ on the interval $[-1, 1]$ using 4 subintervals. Determine the lower sum by choosing $x_i^*$ such that $f(x_i^*)$ is minimized within each subinterval, and the upper sum by choosing $x_i^*$ such that $f(x_i^*)$ is maximized within each subinterval.

In this problem, we're diving into the concept of Riemann sums, a fundamental idea in understanding integration. The purpose here is to approximate the area under a curve, or more precisely, under the function $f(x) = 1 + x^2$ on the interval from -1 to 1. By understanding how to partition an interval and choose sample points, we gain insight into how integrals function as limits of sums. The Riemann sum is a technique that leads us directly into the definition of the Riemann integral.

When working with Riemann sums, we explore two key types: the lower sum and the upper sum, which provide estimates for the area from below and above, respectively. This problem is an exercise in understanding how these sums bound the integral and facilitate its calculation. Specifically, by choosing points within each subinterval such that the function's value is minimized or maximized, we can observe how these choices affect the approximation. This concept ties into understanding how randomness or systematic sampling can affect approximation techniques in numerical integration methods.

Moreover, this type of problem enhances our appreciation of precision in mathematical assumptions and the development of integral calculus. By focusing on a finite number of subintervals, we get a sense of how adding infinitely many intervals bridges the gap between approximation and exact integration. Ultimately, Riemann sums help build a foundation for more advanced topics such as definite integrals, further highlighting their significance in real analysis.