MATH 2B Chapter Notes - Chapter 5.2: Riemann Sum, Riemann Integral, Integral Symbol
14 Dec 2016
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5.2 Definite Integrals
While the last section discussed area, the algebraic constructionit contained is universal.
Definition. Suppose that f is a function defined on an interval [a,b]. Let n be a positive integer, define
∆x=b−a
n, and let
xi=a+i∆x=a+b−a
ni,for each i =0, 1, . . . , n
Choose sample points x∗
i∈[xi−1,xi]. A Riemann Sum is any expression of the form
n
∑
i=1
f(x∗
i)∆x
We say that the function f is Riemann Integrable on [a,b]if
lim
n→∞
n
∑
i=1
f(x∗
i)∆x
converges to the same value for every choice of sample points. In such a case the definite integral of ffrom
ato b is1
Zb
af(x)dx=lim
n→∞
n
∑
i=1
f(xi)∆x
The integral sign Ris a stylized Sto remind you of the word ”Σum.”
f(x)
a=x0x1x2x3x4x5x6b=x7
x
x∗
1x∗
2x∗
3x∗
4x∗
5x∗
6x∗
7
The picture shows a choice of seven sample points. The sum of the (net) areas of the rectangles is a
Riemann sum: if a rectangle is beneath the x-axis, then its area counts negatively.
Theorem. If f is continuous on [a,b], or has only a finite number of jump discontinuities, then f is Riemann
integrable on [a,b].
1Since the limit is the same for all sample points x∗
iwe might as well take right endpoints x∗
i=xi
1
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