Lecture 11: Riemann Sums and Definite Integrals

Lecture 11: Riemann Sums and Definite Integrals
Definition 1 (Riemann Sums)
Let f be defined on [a,b] and let ∆ be a partition of [a,b] given by
a = x 0 < x 1 < x 2 < ... < x n−1 < x n = b
n∑
where ∆x i = x i − x i−1 and c i ∈ [x i−1 ,x i ] then f(c i )∆x i is called the Riemann sum of
f for the partition ∆.
Note that: If every subinterval is of equal width, the partition is regular and the norm
is defined by ‖∆‖ = ∆x = b − a
n .
Definition 2 (Definite Integrals)
n∑
If f be defined on [a,b] and the limit lim f(c i )∆x i exist, then f is integrable on
‖∆‖→0
i=1
[a,b] and the limit is denoted by
n∑
∫ b
lim f(c i )∆x i = f(x)dx.
‖∆‖→0
i=1
a
The limit is called the definite integral of f from a to b.
Theorem 1 If a function f is continuous on [a,b] then f is integrable [a,b].
Example 1 Evaluate the definite integral
∫ 1
−2
i=1
2x dx.
Sol: Let f(x) = 2x and ∆x = b − a
n
= 1 − (−2)
n
∫ 1
n∑
2x dx = lim f(c i )∆x i = lim
−2
‖∆‖→0
i=1
n∑
6 ( 3i
= n→∞
lim − 2 +
n
i=1
n
( 9 )
= n→∞
lim − 12 + 9 + = −3.
n
Note that: The summation formulas:
n∑
n∑ n(n + 1)
n∑
c = cn i = i 2 =
2
i=1
i=1
) 6 = n→∞ lim
n
i=1
= 3 n . Choosing c i = a+i∆x = −2+ 3i
n .
n ∑
n→∞
i=1
(
− 2n +
3
n
n(n + 1)(2n + 1)
6
2 ( − 2 + 3i ) 3 ×
n n
n(n + 1) )
2
n∑
i=1
i 3 = [ n(n + 1) ] 2
2
Exercise:
Evaluate the definite integral by the Riemann sums:
19
∫ 1
0
x 2 dx and
∫ 1
−1
x 3 dx.

Theorem 2 If f is integrable [a,b] and c ∈ [a,b], then
†
∫ a
a
f(x)dx = 0 †
∫ b
a
∫ a ∫ b ∫ c ∫ b
f(x)dx = − f(x)dx † f(x)dx = f(x)dx+ f(x)dx
b
a
a
c
Theorem 3 If f and g are integrable [a,b] and k is a constant then the function of kf
and f ± g are integrable on [a,b]
†
∫ b
a
∫ b
kf(x)dx = k f(x)dx ‡
a
∫ b
a
[ ] ∫ b
f(x) ± g(x) dx = f(x)dx ±
Theorem 4 † If f is integrable and f(x) ≥ 0 on [a,b] then
‡ If f and g are integrable and f(x) ≤ g(x) on [a,b] then
a
∫ b
a
∫ b
a
∫ b
a
f(x)dx ≥ 0.
f(x)dx ≤
g(x)dx
Theorem 5 (Fundamental Theorem of Calculus)
† If f is continuous on [a,b] and F(x) is an antiderivative of f on [a,b] then
∫ b
a
f(x)dx = F(b) − F(a).
‡ If f is continuous on [a,b] then for every x ∈ [a,b],
Example 2 Evaluate the definite integral
Example 3 Evaluate d
dx
∫ x
0
√
t2 + 1 dt.
∫ 2
1
(x 2 − 3)dx and
Theorem 6 (Mean Value Theorem for Integrals)
If f is continuous on [a,b] then there exists c ∈ [a,b] such that
Theorem 7 (Average Value of Function)
If f is integrable on [a,b] then the average value of f on [a,b] is
Exercise:
1. Evaluate:
∫ 4
1
3 √ x dx,
2. Evaluate the derivative:
∫ π/4
0
∫ x
−1
sec 2 x dx,
√
t4 + 1 dt,
∫ 4
u − 2
√ u
du,
1
∫ x+2
3. Find the average value of f(x) = 3x 2 − 2x on [1, 4].
20
x
∫ b
∫
d x
f(t)dt = f(x).
dx a
∫ 3
0
(4t + 1) dt,
∫ 2
0
∫ b
a
|2x − 1| dx.
a
g(x)dx.
f(x)dx = f(c)(b − a).
∫
1 b
f(x)dx.
b − a a
|x 2 −4| dx,
∫ x 3
0
∫ 1
0
sin t 2 dt.
(2t−1) 2 dt.