Lesson 24: The Definite Integral (Section 10 version)
Section 5.2
The Definite Integral
V63.0121, Calculus I
April 15, 2009
Announcements
My office is now WWH 624
Final Exam Friday, May 8, 2:00–3:50pm
. . . . . .
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
Cavalieri’s method in general
Let f be a positive function defined on the interval [a, b]. We want
to find the area between x = a, x = b, y = 0, and y = f(x).
For each positive integer n, divide up the interval into n pieces.
b−a
. For each i between 1 and n, let xi be the ith
Then ∆x =
n
step between a and b. So
x0 = a
b−a
x1 = x0 + ∆x = a +
n
b−a
x2 = x1 + ∆x = a + 2 · …
n
b−a
xi = a + i · …
n
b−a
xn = a + n · =b
n
.. . .. x
.
. 0 . 1 . . . . i . . .. n−1. n
x x. x. x x
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. . . x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. . . . x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. . . . . x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. . . . . . x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
……. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
…….. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
……… x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
………. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
……….. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
………… x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
…………. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
………….. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
…………… x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
……………. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
…………….. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
……………… x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
………………. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
……………….. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
…………………
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
………………….
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
…………………..
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
……………………
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
…………………….
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
……………………..
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
………………………
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
……………………….
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
………………………..
. x
.
matter what choice of ci we
made.
. . . . . .
Theorem of the (previous) Day
Theorem
If f is a continuous function on
[a, b] or has finitely many jump
discontinuities, then
{n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
…………………………
. x
.
matter what choice of ci we
made.
. . . . . .
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
The definite integral as a limit
Definition
If f is a function defined on [a, b], the definite integral of f from a
to b is the number
∫b n
∑
f(x) dx = lim f(ci ) ∆x
∆x→0
a i=1
. . . . . .
Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b
the upper limit)
. . . . . .
Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b
the upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)
. . . . . .
Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b
the upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)
The process of computing an integral is called integration or
quadrature
. . . . . .
The limit can be simplified
Theorem
If f is continuous on [a, b] or if f has only finitely many jump
discontinuities, then f is integrable on [a, b]; that is, the definite
∫b
integral f(x) dx exists.
a
. . . . . .
The limit can be simplified
Theorem
If f is continuous on [a, b] or if f has only finitely many jump
discontinuities, then f is integrable on [a, b]; that is, the definite
∫b
integral f(x) dx exists.
a
Theorem
If f is integrable on [a, b] then
∫ n
∑
b
f(x) dx = lim f(xi )∆x,
n→∞
a i=1
where
b−a
and xi = a + i ∆x
∆x =
n
. . . . . .
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
Estimating the Definite Integral
Given a partition of [a, b] into n pieces, let ¯i be the midpoint of
x
[xi−1 , xi ]. Define
n
∑
Mn = f(¯i ) ∆x.
x
i=1
. . . . . .
Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
. . . . . .
Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
Solution
1 1 3
The partition is 0 < < < < 1, so the estimate is
4 2 4
( )
1 4 4 4 4
M4 = + + +
2 2 2 1 + (7/8)2
4 1 + (1/8) 1 + (3/8) 1 + (5/8)
. . . . . .
Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
Solution
1 1 3
The partition is 0 < < < < 1, so the estimate is
4 2 4
( )
1 4 4 4 4
M4 = + + +
4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2
( )
1 4 4 4 4
= + + +
4 65/64 73/64 89/64 113/64
. . . . . .
Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
Solution
1 1 3
The partition is 0 < < < < 1, so the estimate is
4 2 4
( )
1 4 4 4 4
M4 = + + +
4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2
( )
1 4 4 4 4
= + + +
4 65/64 73/64 89/64 113/64
150, 166, 784
≈ 3.1468
=
47, 720, 465
. . . . . .
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫b
c dx = c(b − a)
1.
a
. . . . . .
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫b
c dx = c(b − a)
1.
a
∫ ∫ ∫
b b b
[f(x) + g(x)] dx = f(x) dx + g(x) dx.
2.
a a a
. . . . . .
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫b
c dx = c(b − a)
1.
a
∫ ∫ ∫
b b b
[f(x) + g(x)] dx = f(x) dx + g(x) dx.
2.
a a a
∫ ∫
b b
cf(x) dx = c f(x) dx.
3.
a a
. . . . . .
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant.
Then
∫b
c dx = c(b − a)
1.
a
∫ ∫ ∫
b b b
[f(x) + g(x)] dx = f(x) dx + g(x) dx.
2.
a a a
∫ ∫
b b
cf(x) dx = c f(x) dx.
3.
a a
∫ ∫ ∫
b b b
[f(x) − g(x)] dx = f(x) dx − g(x) dx.
4.
a a a
. . . . . .
Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f(x) ≥ 0 for all x in [a, b], then
∫ b
f(x) dx ≥ 0
a
7. If f(x) ≥ g(x) for all x in [a, b], then
∫ ∫
b b
f(x) dx ≥ g(x) dx
a a
. . . . . .
Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f(x) ≥ 0 for all x in [a, b], then
∫ b
f(x) dx ≥ 0
a
7. If f(x) ≥ g(x) for all x in [a, b], then
∫ ∫
b b
f(x) dx ≥ g(x) dx
a a
8. If m ≤ f(x) ≤ M for all x in [a, b], then
∫ b
m(b − a) ≤ f(x) dx ≤ M(b − a)
a
. . . . . .
Example
∫ 2
1
dx using the comparison properties.
Estimate
x
1
. . . . . .
Example
∫ 2
1
dx using the comparison properties.
Estimate
x
1
Solution
Since
1 1
≤x≤
2 1
for all x in [1, 2], we have
∫ 2
1 1
·1≤ dx ≤ 1 · 1
x
2 1
. . . . . .