Chapter 5 – The Definite Integral

5.1 Estimating with Finite SumsExample Finding Distance

Traveled when Velocity Varies2A particle starts at 0 and moves along the -axis with velocity ( )

for time 0. Where is the particle at 3?

x x v t t

t t

Graph and partition the time interval into subintervals of length . If you use

1/ 4, you will have 12 subintervals. The area of each rectangle approximates

the distance traveled over the subint

v t

t

erval. Adding all of the areas (distances)

gives an approximation to the total area under the curve (total distance traveled)

from 0 to 3.t t

2

Continuing in this manner, derive the area 1/ 4 for each subinterval and

add them:

1 9 25 49 81 121 169 225 289 361 441 529 2300

256 256 256 256 256 256 256 256 256 256 256 256 2568.98

im

LRAM, MRAM, and RRAM approximations to the area under the graph of y=x2 from x=0 to x=3

p.270 (1-19, 26, 27)

5.2 Definite Integrals1 2 3 11

...n

k n nka a a a a a

5

1kk

3

11k

kk

2

1 1k

k

k 25

4 1k

k

k

Sigma notation enables us to express a large sum in compact form:

Ex)

Ex) Ex)

Ex)

The Definite Integral as a Limit of Riemann Sums

-1

0

Let be a function defined on a closed interval [ , ]. For any partition

of [ , ], let the numbers be chosen arbitrarily in the subinterval [ , ].

If there exists a number such that lim

k k k

P

f a b P

a b c x x

I 1

( )

no matter how and the 's are chosen, then is on [ , ] and

is the of over [ , ].

n

k kk

k

f c x I

P c f a b

I f a b

integrable

definite integral

All continuous functions are integrable. That is, if a function is

continuous on an interval [ , ], then its definite integral over

[ , ] exists.

f

a b

a b

We have that 1

limbn

k kn

k a

f c x f x dx

b

a

f x dxUpper limit

Integral sign

Lower limit

Integrand

Variable of Integration

th

2

1

The interval [-2,4] is partitioned into subintervals of equal length 6 / .

Let denote the midpoint of the subinterval. Express the limit

lim 3 2 5 as an integral.

k

n

k kn k

n x n

m k

m m x

Example Using the Notation

Area Under a Curve

If ( ) is nonnegative and integrable over a closed interval [ , ],

then the area under the curve ( ) from to is the

, ( ) .b

a

y f x a b

y f x a b

A f x dx

integral

of from to f a b

Area= ( ) when ( ) 0.

( ) area above the -axis area below the -axis .

b

a

b

a

f x dx f x

f x dx x x

Notes about Area

The Integral of a Constant

If ( ) , where is a constant, on the interval [ , ], then

( ) ( ) b b

a a

f x c c a b

f x dx cdx c b a

2

-1Evaluate numerically. sinx xdx

Evaluate the following integrals:2

2

2

4 x dx

2

1

xdx

x

FNINT( sin , ,-1,2) 2.04x x x

p.282 (1-27, 33-39) odd

5.3 Definite Integrals and

Antiderivatives

1 4 1

-1 1 -1

1

4

Suppose ( ) 5, ( ) 2, and ( ) 7 .

Find ( ) if possible.

f x dx f x dx h x dx

f x dx

1 4 1

-1 1 -1

4

1

Suppose ( ) 5, ( ) 2, and ( ) 7 .

Find ( ) if possible.

f x dx f x dx h x dx

f x dx

1 4 1

-1 1 -1

2

2

Suppose ( ) 5, ( ) 2, and ( ) 7 .

Find ( ) if possible.

f x dx f x dx h x dx

h x dx

Ex: Show that the value of 1

0

31 cos

2xdx

Average (Mean) Value

If is integrable on [ , ], its average (mean) value on [ , ] is

1( ) ( )b

a

f a b a b

avg f f x dxb a

2Find the average value of ( ) 2 on [0,4].f x x

The Mean Value Theorem for Definite Integrals

If is continuous on [ , ], then at some point in [ , ],

1( ) ( ) .b

a

f a b c a b

f c f x dxb a

1, 1

1

1

nn xx dx C n

n

dx dx x C

cossin

kxkxdx C

k

sincos

kxkxdx C

k

2sec tanxdx x C

2csc cotxdx x C

sec tan secx xdx x C csc cot cscx xdx x C

Integral Formulas

This is known as the indefinite integral. C is a constant.

5x dx 1dxx

sin 2xdxcos

2

xdx

Evaluate:

p. 290 (1 – 29) odd19 – 29 noteDo (31-35)After 5.4

5.4 Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus – Part 1

If is continuous on [ , ], then the function ( ) ( )

has a derivative at every point in [ , ], and

( ) ( ).

x

a

x

a

f a b F x f t dt

x a b

dF df t dt f x

dt dx

Evaluate the following:

cosxd

tdtdx

2

0

1

1

xddt

dx t

2

1

cosx

y tdt

Find dy

dx

Find dy

dx

5

3 sinx

y t tdt2

2

1

2

x

tx

y dte

Find a function y = f(x) with derivative

That satisfies the condition f(3) = 5.

tandy

xdx

The Fundamental Theorem of Calculus, Part 2

If is continuous at every point of [ , ], and if is any antiderivative

of on [ , ], then ( ) ( ) - ( ).

This part of the Fundamental Theorem is also called the

.

b

a

f a b F

f a b f x dx F b F a Integral

Evaluation Theorem

3 2

-1Evaluate 3 1 using an antiderivative.x dx

How to Find Total Area AnalyticallyTo find the area between the graph of ( ) and the -axis over the interval

[ , ] analytically,

1. partition [ , ] with the zeros of ,

2. integrate over each subinterval,

3. add the absolute values o

y f x x

a b

a b f

f

f the integrals.

Find the area of the region between the curve y = 4 – x2, [0, 3] and the x-axis.

Look at page 301 example 8.

p.302 (1-57) odd

5.5 Trapezoidal Rule

0 1 11 2

0

1 2 1

0 1 2 1

0 1 1 1 1

( ) ...2 2 2

...2 2

2 2 ... 2 ,2

where ( ), ( ), ..., ( ), ( ).

b n n

a

n

n

n n

n n n

y y y yy yf x dx h h h

y yh y y y

hy y y y y

y f a y f x y f x y f b

The Trapezoidal Rule

0 1 2 1

To approximate ( ) , use

2 2 ... 2 ,2

where [ , ] is partitioned into n subintervals of equal length

( - ) / .

LRAM RRAMEquivalently, ,

2where LRAM and RRAM are the Rienamm

b

a

n n

n n

n n

f x dx

hT y y y y y

a b

h b a n

T

sums using the left

and right endpoints, respectively, for for the partition.f

Use the trapezoidal rule with n = 4 to estimate . Compare with fnint.

Ex: An observer measures the outside temperature every hour from noon until midnight, recording the temperatures in the following table.

What was the average temperature for the 12-hour period?

22

1

x dx

Time N 1 2 3 4 5 6 7 8 9 10 11 M

Temp 63 65 66 68 70 69 68 68 65 64 62 58 55

Simpson’s Rule

0 1 2 3 2 1

To approximate ( ) , use

4 2 4 ... 2 4 ,3

where [ , ] is partitioned into an even number subintervals

of equal length ( - ) / .

b

a

n n n

f x dx

hS y y y y y y y

a b n

h b a n

Ex: Use Simpson’s rule with n = 4 to approximate 2

4

0

5x dx

p.312 (1-18)