MAT 21B Lecture Notes - Lecture 4: Riemann Sum, Riemann Integral, Royal Institute Of Technology
5.3: The Definite Integral
Last time, we defined Riemann sums as follows.
Let P={x0, x1,· · · , xn}be any partition of [a, b], and let ckbe a point in the kth subinterval [xk−1, xk], the
corresponding Riemann sum is defined by
SP=
n
X
k=1
f(ck)·∆xk.
This is called a Riemann Sum for fon [a, b]. It approximates the “signed area” of the region R.
Limits of Riemann Sums: If the values of all Riemann sums approach a limiting value I, as the norm of
partitions approach zero,
lim
||P||→0
SP= lim
||P||→0
n
X
k=1
f(ck)∆xk=I
we call this value I the definite integral of f.
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MAT 21B Full Course Notes
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Document Summary
Last time, we de ned riemann sums as follows. Let p = {x0, x1, , xn} be any partition of [a, b], and let ck be a point in the kth subinterval [xk 1, xk], the corresponding riemann sum is de ned by. This is called a riemann sum for f on [a, b]. It approximates the signed area of the region r. Limits of riemann sums: if the values of all riemann sums approach a limiting value i, as the norm of partitions approach zero, lim. Xk=1 f (ck) xk = i we call this value i the de nite integral of f. The de nite integral: the symbol for the number i is. When the limit exists, we say f is (riemann) integrable over [a,b], and the limit value is called the (de nite) integral of f over [a, b]. Question: which functions are (riemann) integrable? i. e. for which functions, the de nite integral always lim.
