Upper and lower sums are mathematical concepts used to approximate the area under a curve or the value of a function. They are calculated by dividing the curve into smaller intervals and finding the maximum and minimum values within each interval, respectively.
The purpose of calculating upper and lower sums is to estimate the exact value of a function or the area under a curve, especially when the function or curve is difficult to integrate. They provide a closer approximation to the actual value as the number of intervals increases.
Upper and lower sums differ in the way they are calculated. Upper sums use the maximum value of each interval to approximate the area, while lower sums use the minimum value. This results in an overestimation and underestimation, respectively, of the actual value.
The relationship between upper and lower sums is that as the number of intervals decreases, the upper sum will approach the lower sum and both will approach the actual value. This is known as the squeeze theorem and is used to prove the convergence of upper and lower sums to the exact value.
Upper and lower sums are used in real-life applications to approximate the area under a curve in various fields such as engineering, physics, economics, and statistics. They are also used in financial calculations, such as estimating the profit or loss of an investment over a period of time.