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

. . . . . .

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

. . . . . .