Follow the link to the slide.

Then click on the figure to play the animation.A

Figure 5.1.8 Figure 5.1.9

Section 5.1 Figures 8, 9Upper and lower estimates of the area in Example 2

A

A

Section 1 / Figure 1

Section 5.1 Figure 12Approximations of the area for n = 2, 4, 8 and 12 rectangles

2)a( n 4)b( n

8)c( n 12)d( n

Section 5.1 Figure 13Approximating rectangles when sample points are not endpoints

5. 1 Sigma notation

4

1

1 2 3 4 163

1 2 3 4 5 60k

k

k

5

1

1 2 3 4 5k

k

3

1 1 2 2 3 31

( ) ( ) ( ) ( )k kk

f x x f x x f x x f x x

36

1

1 8 27 64 125 216i

i

The graph of a typical function y = ƒ(x) over [a, b]. Partition [a, b] into n subintervals a < x1 < x2 <…xn < b. Select a number in each subinterval ck. Form the product f(ck)xk. Then take the sum of these products.

1

( )n

k kk

f c x

5.2

The curve of with rectangles from finer partitions of [a, b]. Finer partitions create more rectangles with shorter bases.

01

lim ( )n

k kPk

f c x

1

( )n

k kk

f c x

01

lim ( )n

k kPk

f c x

The definite integral of f(x) on [a, b]

( )b

a

f x dxIf f(x) is non-negative, then the definite integral represents the area of the region under the curve and above the x-axis between the vertical lines x =a and x = b

Rules for definite integrals

( ) ( )b a

a b

f x dx f x dx

( ) 0a

a

f x dx

( ) ( )b b

a a

kf x dx k f x dx

( ( ) ( )) ( ) ( )a a a

b b b

f x g x dx f x dx g x dx

( ( )) ( ) ( )b c b

a a c

f x dx f x dx f x dx

5.3 The Fundamental Theorem of Calculus

1. ( ) ( ) , ( ) ( )x

a

If g x f t dt then g x f x Where f(x) is continuous on [a,b] and differentiable on (a,b)

( ) ( )x

a

df t dt f x

dx

Find the derivative of the function:3( ) ( 5 sin )

x

a

g x t t t dt

The Fundamental Theorem of Calculus

2. ( ) ( ) ( ) ( ) ( )b

a

f x dx F b F a where F x f x

32

0

sec d

42

2

( 3 )x x dx

Indefinite Integrals

( ) ( ) ( ) ( )f x dx F x means F x f x

Definite Integrals

( )( ) ( ) ( ) ( )b

af x dx F b F a where F x f x

21) (2sin 5cos 3sec )d

3 22)

ydy

y

32

1

3) (1 )x dx

Try These

2

2cos 5sin 3tan

1) (2sin 5cos 3sec )d

C

3 1

1 1

2

2 2

2

3 22) (3 2 )

2 4

ydy y y dy

y y C

y

3

3 32 2 2

1

4

5 3

1 1

3) (1 ) (1 2 )

2 1[ ] 69.6 1.8667 67.733

3 5

x dx x x dx

x x x

Answers

Indefinite Integrals and Net Change

• The integral of a rate of change is the net change.

( ) ( ) ( )b

a

f x dx F b F a net area

| ( ) | totalb

a

f x dx area

If the function is non-negative, net area = area.If the function has negative values, the integral must be split into separate parts determined by f(x) = 0. Integrate one part where f(x) > 0 and the other where f(x) < 0.

but

Indefinite Integrals and Net Change

• The integral of a rate of change is the net change.

2

1

2 1( ) ( ) ( )t

t

v t dt s t s t net change or displacement 2

1

| ( ) | total distance travelledt

t

v t dt

If the function is non-negative, displacement = distance.If the function has negative values, the integral Must be split into separate parts determined by v = 0. Integrate one part where v>0 and the other where v<0.

5.5 Review of Chain rule

2

2 313 1 6 1

3x x x

2

2 3

6 1

3 3 1

x

x x

1

2 33 1d

x xdx

3 2sin 3 sin sin

d d

dx dx 23sin cos

3sind

dx

If F is the antiderivative of f

( ( )) ( ) ( ( ))f g x g x dx F g x C

( ( )) (

( )

) ) ( )

(

(

)

f g x g x d

u g x

du g x

x f u du F u C

dx

( ( ))F g x C

2

2 31(6 1) 3 1

3x x x dx

22 3

6 1

3 3 1

xdx

x x

23 1

(6 1)

u x x

du x dx

1

313

3u C

2

31

3u du

1

2 3(3 1)x x C

Let u = inside function of more complicated factor.

Check by differentiation

23sin cos d sin

cos

u

du d

3u C 23 u du 3sin C

Let u = inside function of more complicated factor.

Check by differentiation

Substitution with definite integrals7

0

4 3xdx4 3

3

3

u x

du dx

dudx

71 3 3 3 3

2 2 2 2 2

0

1 1 2 2 2 234(4 3 ) (25 4 )

3 3 3 9 9 9u du u C x

2525 1 3 3 3

2 2 2 2

4 4

1 2 2 234(25 4 )

3 9 9 9u du u

Using a change in limits

The average value of a function on [a, b]

1( )

b

a

Average f x dxb a