Any standard real analysis course at university introduces the Sandwich Theorem to help us evaluate the limits or values of complicated functions and expressions.
This same reasoning can extend to the realm of integration in that an integral can be approximated by upper and lower Riemann sums.
As you can see in the image above, the left-hand side is the lower sum and the right-hand side is the upper sum.
We are interested in finding the two Riemann sums in order to evaluate the above integral. In fact, for the left-hand graph, the above integral is defined as the supremum of all areas shaded in that graph. Equally, the integral is defined as the infimum of all areas shaded in the right-hand graph.
The approximation we obtain with both the upper and lower sums gets more and more accurate as the number of rectangle strips increases. This means the approximations eventually converge to the true value of the integral.
One way to visualize this is to think of the strips getting thinner and thinner and eventually filling the area under the curve perfectly.
Keep reading with a 7-day free trial
Subscribe to Mathematical Donuts 🍩 to keep reading this post and get 7 days of free access to the full post archives.