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1

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral5.1 The Area ProblemGoal: To find the area under the graph of f(x)

and above the x-axis between x = a and x = b

How:

Problem:

Use rectangles to approximate the area

curved sides

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral( ) [ ]2 on the interval 1,5f x x=

using 4 rectangles

and the endpoint

of each interval to get

the height of each rectangle

right

( ) ( ) ( ) ( )Area 2 1 3 1 4 1 5 1f f f f≈ ⋅ + ⋅ + ⋅ + ⋅

Area 4 9 16 25≈ + + +

2Area 54 units≈

11/13/2013

2

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral( ) [ ]2 on the interval 1,5f x x=

using 4 rectangles

and the endpoint

of each interval to get

the height of each rectangle

left

Area 1 4 9 16≈ + + +

2Area 30 units≈

( ) ( ) ( ) ( )Area 1 1 2 1 3 1 4 1f f f f≈ ⋅ + ⋅ + ⋅ + ⋅

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral( ) [ ]2 on the interval 1,5f x x=

using 4 rectangles

and the

of each interval to get

the height of each rectangle

midpoint

9 25 49 81Area

4 4 4 4≈ + + +

2164Area units

4≈

( ) ( ) ( ) ( )3 5 7 92 2 2 2

Area 1 1 1 1f f f f≈ ⋅ + ⋅ + ⋅ + ⋅

2Area 41 units≈

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Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral( ) [ ]2 on the interval 1,5f x x=

using 8 rectangles

and the endpoint

of each interval to get

the height of each rectangle

right

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral( ) [ ]2 on the interval 1,5f x x=

using 8 rectangles

and the endpoint

of each interval to get

the height of each rectangle

right

1 9 25 49 81Area 4 9 16 25

2 4 4 4 4

≈ + + + + + + +

21 164Area 54 units

2 4

≈ +

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 5 7 91 1 1 1 1 1 1 12 2 2 2 2 2 2 2 2 2 2 2

Area 2 3 4 5f f f f f f f f≈ ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅

295Area units

2≈

2Area 47.5 units≈

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Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

11/13/2013

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Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

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Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

as the number

of rectangles

increases, accuracy

increases

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

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Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral5.3 The Definite Integral

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

( )b

a

f x dx∫

Definite Integral

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Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

* and i i i

x x x a i x= = + ∆

( )*

1

limn

in

i

f x x→∞

=

∆∑

b ax

n

−∆ =

a a x+ ∆ 2a x+ ∆ 3a x+ ∆

�

a i x+ ∆

0x1x 2x 3x

ix

using right endpoints we can simnplify the Riemann sum

( ) ( ) ( ) ( )1 2limi n

nx f x f x f x f x

→∞∆ + + + + + � �

( )*

1

limn

in

i

f x x→∞

=

∆∑ ( )1

limn

b a

nn

i

b af a i

n

−

→∞=

−= + ⋅∑

5.2 Riemann Sum

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

Appendix E : Sigma Notation page A37

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9

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral( ) [ ]2 on the interval 1,5f x x=

Find the exact area using the Riemann sum ( ) ( )1

lim

b n

b a

nn

ia

b af x dx f a i

n

−

→∞=

−= + ⋅∑∫

( )5

2 4

11

4lim 1

n

nn

i

x dx f in→∞

=

= + ⋅∑∫ ( )2

4

1

4lim 1

n

nn

i

in→∞

=

= + ⋅∑ ( )2

2

8 16

1

4lim 1

n

i in nn

in→∞=

= + +∑

2

2

8 16

1 1 1

4lim 1

n n n

i in nn

i i in→∞= = =

= + +

∑ ∑ ∑

11

2

21

4 8li 1

16m

n

i

n

n

n

i ini

ni

n ==→

=∞

= + +

∑∑ ∑

( ) ( )( )2

4 8 1 16li

2

21 1

6m

n n n nn

n nn n n

→∞

+ = + ⋅ + ⋅

+

+ ( ) ( )( )2 3

1 1 2 132 64lim 4

2 6n

n n n n n

n n→∞

+ + + = + ⋅ + ⋅

32lim 4n→∞

= +

16

2n

n⋅

( )1

2

n + 64+

32

3n 2

n⋅

( ) ( )22 3 1

1 2 1

6

n n

n n+ +

+ +3

( )2

2

32 2 3 116 16lim 4

3n

n nn

n n→∞

+ ++ = + +

2 2

644 16

3

16 3 2l

2 3

3im

n n n n→∞

= + + + +

+

0 0 0

644 16

3= + +

60 64

3

+=

124

3=

1= 41

3Too much work. We find an easier way in section 5.3

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

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Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

Not all functions are integrable

the definite integral measures net area

Area under the x-axis is considered negative

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Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

( )1. 0

a

a

f x dx =∫

Properties :

( ) ( )2.

a b

b a

f x dx f x dx= −∫ ∫

( ) ( ) ( ) ( )3.

b b b

a a a

f x g x dx f x dx g x dx± = ± ∫ ∫ ∫

( ) ( )4.

b b

a a

cf x dx c f x dx=∫ ∫

( )5.

b

a

cdx c b a= −∫

( ) ( ) ( )6. If , then

b c b

a a c

a c b f x dx f x dx f x dx< < = +∫ ∫ ∫

Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite Integral

2

2 33

−3 −4

−1

( )2

0

. i f x dx =∫ 4

( )5

0

. ii f x dx =∫ ( ) ( )2 5

0 2

f x dx f x dx+∫ ∫ 4 6= + = 10

( )7

5

. iii f x dx =∫ −3

( )9

0

. iv f x dx =∫ ( ) ( )5 9

0 5

f x dx f x dx+∫ ∫ ( )10 8= + − = 2

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Math 103 – Rimmer

5.1 The Area Problem

5.3 The Definite IntegralEvaluate the integral by interpreting it in terms of areas.

( )5

2

0

1 25 x dx+ −∫21 25y x= + −

only the right upper

quarter circle

( )22

1 25x y+ − =

215

4rπ= +

=25π

+ 54

12

0

6x dx−∫ 6y x= −

shifted 6

units to the right

y x= ( )( )1

6 62

( )( )1

6 62

18 18

= 36

( )center 0,1 , radius 5=

5