Week 6 Lecture Notes, Lec01 7.7. Approximate Integration Sometimes it is impossible to evaluate an integral directly. For example, when the integrand is consisted of elementary functions whose anti-derivatives cannot be found. ∫ ? 𝑥 2 ?𝑥 ; ∫ sin(𝑥 2 ) ?𝑥 ; ∫ ?𝑥 ln 𝑥 ; ∫ √ 1 + 𝑥 3 ?𝑥 ; Or, in practical applications, when some data points are collected and it’s not even possible to find a function which can represent such data in terms of a function. For example, when the rates of changes in something (e.x. population) is measured, and the area under the curve defines the actual function (because it returns the anti-derivative which in this case is population). In these cases, we can approximate the integral using the following methods. ❖ Riemann Sums As it was introduced in Calculus I, the area under a curve for ? ≤ 𝑥 ≤ ? , can be found by dividing [a,b] into smaller sub-intervals and then approximate the area in each interval, then summing them up.