Week 6 - Lec01.pdf - Week 6 Lecture Notes Lec01 7.7 Approximate Integration Sometimes it is impossible to evaluate an integral directly For example when

Week 6 - Lec01.pdf - Week 6 Lecture Notes Lec01 7.7...

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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.
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