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.