aim: what are riemann sums?

Aim: Riemann Sums & Definite Integrals Course: Calculus

Do Now:

Aim: What are Riemann Sums?

Approximate the area under the curve y = 4 – x2 for [-1, 1] using 4 inscribed rectangles.


Devising a Formula•Using left endpoint to approximate area under the curve is

0 1 2 1nb a

y y y yn

4

3.5

3

2.5

2

1.5

1

0.5

1 2

f x = x2

lower sum

xa b

12

yn - 1yn - 2y0

y1

yn - 1

the more rectangles the better the

approximation

the exact area?

take it to the limit!

0 1 2 1lim nn

b a y y y yn

left endpoint formula


•Using right endpoint to approximate area under the curve is

Right Endpoint Formula

4

3.5

3

2.5

2

1.5

1

0.5

1 2

f x = x2

x

upper sum

a b

y0

y1

yn - 1

yn

1 2 3 nb a

y y y yn

1 2 3lim nn

b a y y y yn

right endpoint formula

1 3 5 2 12 2 2 2

lim nn

b ay y y y

n

midpoint formula


Sigma Notation

sigma

sum of terms

The sum of the first n terms of a sequence is represented by

n

i 1 2 3 4 n,i 1

a a a a a aL

where i is the index of summation,n is the upper limit of summation, and1 is the lower limit of summation.

99

i 1i 1 2 3 99

L


Summation Formulas

constantanyis,.111

caccan

ii

n

ii

n

i

n

iii

n

iii baba

1 11)(.2

cncn

i

1.3

2)1(...321.4

1

nnni

n

i

6)12)(1(...21.5 222

1

2

nnnni

n

i

4)1(...321.6

223333

1

3

nnni

n

i


Riemann Sums•A function f is defined on a closed interval [a, b].•It may have both positive and negative values on the interval.•Does not need to be continuous.

Δx1 Δx2 Δx3 Δx4 Δx5 Δx6

a = = bx0 x6x1 x2 x3 x4 x5

1x 2x 3x 4x 5x 6x

1

0.5

-0.5

-1

1 2 3

Partition the interval into n subintervals not necessarily of equal length.

a = x0 < x1 < x2 < . . . < xn – 1 < xn = b

- arbitrary/sample points for ith intervalix

Δxi = xi – xi – 1


Riemann Sums

Partition interval into n subintervals not necessarily of equal length.

Δx1 Δx2 Δx3 Δx4 Δx5 Δx6

a = x0 x6 = bx1 x2 x3 x4 x5

1x 2x 3x 4x 5x 6x

- arbitrary/sample points for ith intervalix

1

Riemann Sumn

iP ii

R f x x

0 1 2 1nb a b a b a b a

y y y yn n n n

ci = xi


Riemann Sums

1

Riemann Sumn

iP ii

R f x x

Δx1 Δx2

Δx6

x6 = ba = x0

1x 2x 5x

Δx4

Δxi = xi – xi – 1


Definition of Riemann Sum

Let f be defined on the closed interval [a, b], and let Δ be a partition of [a, b] given by

a = x0 < x1 < x2 < . . . . < xn – 1 < xn = b,

where Δxi is the length of the ith subinterval. If ci is any point in the ith subinterval, then the sum

is called a Riemann sum for f for the partition Δ

11

( ) n

i i i i ii

f c x x c x

largest subinterval – norm - ||Δ|| or |P|

b a

n

equal subintervals – partition is regularb a

xn

regular partition general partition

0 implies n

converse not true implies 0n


Model ProblemEvaluate the Riemann Sum RP for

f(x) = (x + 1)(x – 2)(x – 4) = x3 – 5x2 + 2x + 8 on the interval [0, 5] using the Partition P with partition points 0 < 1.1 < 2 < 3.2 < 4 < 5 and corresponding sample points

1 2 3 4 50.5, 1.5, 2.5, 3.6, 5x x x x x

5

1

1 2 31 2 3

4 54 5

iP ii

R f x x

f x x f x x f x x

f x x f x x

1

Riemann Sumn

iP ii

R f x x


Model Problem

5

1

0.5 1.1 0 1.5 2 1.1

2.5 3.2 2 3.6 4 3.2 5 5 4

iP ii

R f x x

f f

f f f

7.875 1.1 3.125 0.9 2.625 1.2

2.944 0.8 18 1 23.9698


Definition of Definite Integral

If f is defined on the closed interval [a, b] and the limit

exists, the f is integrable on [a, b] and the limit is denoted by

The limit is called the definite integral of f from a to b. The number a is the lower limit of integration, and the number b is the upper limit of integration.

0 1lim ( )

n

i ii

f c x

0 1

lim ( ) ( )n b

i i ai

f c x f x dx

Definite integral is a numberIndefinite integral is a family of functions

If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].

1

limn

i in i

f c x


Evaluating a Definite Integral as a Limit

2

1

-1

-2

-3

f x = 2x

1

2Evaluate the definite integral 2x dx

3

ib a

x xn n

32ii

c a i xn

1

2 0 1

2 limn

i ii

x dx f c x

1

limn

i in i

f c x

1

3 3lim 2 2n

n i

in n

1

6 3lim 2n

n i

in n

ci = xi


Evaluating a Definite Integral as a Limit1

21

6 32 lim 2n

n i

ix dx

n n

16 3lim 22n

n nn

n n

9lim 12 9 3n n

The Definite Integral as Area of Region

If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded

by the graph of f, the x-axis and the vertical lines x = a and x = b is given by

Area = b

af x dx

2

1

-1

-2

-3

f x = 2x

not the area

cncn

i

1.3

1

( 1)4.2

n

i

n ni


Properties of Definite Integrals

1. If is defined at , then 0

2. If is integrable on [ , ],

then

If is integrable on the three closed intervalsdetermined by , , and , then

3.

a

a

a b

b a

b c c

a a b

f x a f x dx

f a b

f x dx f x dx

fa b c

f x dx f x dx f x dx

If and are integrable on [ , ] and is constant, then the functions of and are integrable on [ , ], and

4.

5. ( )

b b

a a

b b b

a a a

f g a b kkf f g

a b

kf x dx k f x dx

f x g x dx f x dx g x dx


6

4

2

5

g x = x+2

Areas of Common Geometric Figures

2 2

24 x dx

2

Sketch & evaluate area region using geo. formulas.3

14 dx

3

02x dx= 8

1 212

A b b

212

2A r

2

6

4

2

5

f x = 4

A = lw

3

11

1

4 lim

3 1lim 4

2lim 4

lim 4 2 8

n

i in i

n

n i

n

n

dx f c x

n

nn


Model Problems

0

3

Evaluate sin

2

x dx

x dx

=0

3

02x dx

212

6

4

2

5

g x = x+2

2 2

1Evaluate ( 1)x dx


Model Problem

3 2

1

3 3 32

1 1 1

Evaluate 4 3 using each

of the following values.26 , 4, 23

x x dx

x dx x dx dx

3 2

1

3 3 32

1 1 1

4 3

4 3

x x

x dx x dx dx

3 3 32

1 1 1= 4 3x dx x dx dx

26 4= 4 4 3 23 3


Model Problem

3 2

1Evaluate 2 8x dx

15

10

5

-5

-10

2 4

f x = 2x2-8

2

1lim 2 8

n

in i

x x

A1

A2

Total Area = -A1 + A2

41 1ii

c i xn

0 1

lim ( )n

i ii

f c x

4i

b ax x

n n

2

1

4 42 1 8n

i

in n


Model Problem15

10

5

-5

-10

2 4

f x = 2x2-8

A1

A2

2

1

4 42 1 8n

i

in n

2

21

8 16 42 1 8n

i

i in n n

2

21

16 32 46n

i

i in n n

2

2 31

24 64 128n

i

i in n n

2

2 31 1 1

24 64 128n n n

i i i

i in n n


Model Problem

2

2 31 1 1

24 64 128n n n

i i i

i in n n

22 3

1 1 1

24 64 1281n n n

i i i

i in n n

2

1 128 3 124 32 1 26n n n

2

1 128 3 1lim 24 32 1 26n n n n

128 4024 323 3

take the limit n


Definition of Riemann Sum


Subintervals of Unequal Lengths

( ) for the -axis for 0 1f x x x x

1

lim ( )n

i in i

f c x

2

1 2 and is the

width of the th interval

ix x

ni

22

1 2 2

1iix

n n

1.4

1.2

1

0.8

0.6

0.4

0.2

0.5 1

1nn

1n

2n

2

1n

22

2 2

12 nn n

2 2

1 2

2 1i i ix

n

1 2

2 1ix

n


Subintervals of Unequal Lengths

( ) for the -axis for 0 1f x x x x

1

lim ( )n

i in i

f c x

1.4

1.2

1

0.8

0.6

0.4

0.2

0.5 1

1 2

2 1ix

n

2

2 21

2 1limn

n i

i in n

23

1

1lim 2n

n i

i in

3

1 2 1 11lim 26 2n

n n n n nn

3 2

31

4 3 2lim ( ) lim6 3

n

i in ni

n n nf c x

n

aim: what are riemann sums?

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