11x1 t14 01 area under curves

IntegrationArea Under Curve

IntegrationArea Under Curve

IntegrationArea Under Curve

IntegrationArea Under Curve

33 2111Area

IntegrationArea Under Curve

33 2111Area

9Area

IntegrationArea Under Curve

33 2111Area

9Area

IntegrationArea Under Curve

33 2111Area

9Area

33 1101

IntegrationArea Under Curve

33 2111Area

9Area

33 1101

1

IntegrationArea Under Curve

33 2111Area

9Area

33 1101

1

2unit5AreaEstimate

IntegrationArea Under Curve

33 2111Area

9Area

33 1101

1

2unit5AreaEstimate

24 unitExact Area

33333 26.12.18.04.04.0Area

33333 26.12.18.04.04.0Area

76.5Area

33333 26.12.18.04.04.0Area

76.5Area

33333 26.12.18.04.04.0Area

76.5Area 33333 6.12.18.04.004.0

33333 26.12.18.04.04.0Area

76.5Area 33333 6.12.18.04.004.0

56.2

33333 26.12.18.04.04.0Area

76.5Area 33333 6.12.18.04.004.0

56.22unit4.16AreaEstimate

3333333333 28.16.14.12.118.06.04.02.02.0Area

3333333333 28.16.14.12.118.06.04.02.02.0Area 84.4Area

3333333333 28.16.14.12.118.06.04.02.02.0Area 84.4Area

3333333333 28.16.14.12.118.06.04.02.02.0Area 84.4Area

3333333333 8.16.14.12.118.06.04.02.002.0

3333333333 28.16.14.12.118.06.04.02.02.0Area 84.4Area

3333333333 8.16.14.12.118.06.04.02.002.0

24.3

3333333333 28.16.14.12.118.06.04.02.02.0Area 84.4Area

3333333333 8.16.14.12.118.06.04.02.002.0

24.32unit4.04AreaEstimate

As the widths decrease, the estimate becomes more accurate, lets investigate one of these rectangles.

y

x

y = f(x)

As the widths decrease, the estimate becomes more accurate, lets investigate one of these rectangles.

y

x

y = f(x)

As the widths decrease, the estimate becomes more accurate, lets investigate one of these rectangles.

y

x

y = f(x)

A(c) is the area from 0 to c

c

As the widths decrease, the estimate becomes more accurate, lets investigate one of these rectangles.

y

x

y = f(x)

A(c) is the area from 0 to c

c

A(x) is the area from 0 to x

x

A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle;

A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle;

f(x)

x - c

A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle;

f(x)

x - c

xfcxcAxA

A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle;

f(x)

x - c

xfcxcAxA

cx

cAxAxf

A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle;

f(x)

x - c

xfcxcAxA

cx

cAxAxf

h

cAhcA h = width of rectangle

A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle;

f(x)

x - c

xfcxcAxA

cx

cAxAxf

h

cAhcA h = width of rectangle

As the width of the rectangle decreases, the estimate becomes more accurate.

exact becomesAreathe,0 as i.e. h

exact becomesAreathe,0 as i.e. h

h

cAhcAxfh

0lim

exact becomesAreathe,0 as i.e. h

h

cAhcAxfh

0lim

h

xAhxAh

0lim xch ,0as

exact becomesAreathe,0 as i.e. h

h

cAhcAxfh

0lim

h

xAhxAh

0lim xch ,0as

xA

exact becomesAreathe,0 as i.e. h

h

cAhcAxfh

0lim

h

xAhxAh

0lim xch ,0as

xA

function.Areatheofderivativetheiscurvetheofequation the

exact becomesAreathe,0 as i.e. h

h

cAhcAxfh

0lim

h

xAhxAh

0lim xch ,0as

xA

function.Areatheofderivativetheiscurvetheofequation the

is;andbetween curveunder the areaThe bxaxxfy

exact becomesAreathe,0 as i.e. h

h

cAhcAxfh

0lim

h

xAhxAh

0lim xch ,0as

xA

function.Areatheofderivativetheiscurvetheofequation the

is;andbetween curveunder the areaThe bxaxxfy

b

a

dxxfA

exact becomesAreathe,0 as i.e. h

h

cAhcAxfh

0lim

h

xAhxAh

0lim xch ,0as

xA

function.Areatheofderivativetheiscurvetheofequation the

is;andbetween curveunder the areaThe bxaxxfy

b

a

dxxfA

aFbF

exact becomesAreathe,0 as i.e. h

h

cAhcAxfh

0lim

h

xAhxAh

0lim xch ,0as

xA

function.Areatheofderivativetheiscurvetheofequation the

is;andbetween curveunder the areaThe bxaxxfy

b

a

dxxfA

aFbF

xfxF offunction primitivetheiswhere

e.g. (i) Find the area under the curve , between x = 0 and x= 2

3xy

e.g. (i) Find the area under the curve , between x = 0 and x= 2

3xy

2

0

3dxxA

e.g. (i) Find the area under the curve , between x = 0 and x= 2

3xy

2

0

3dxxA2

0

4

41

x

e.g. (i) Find the area under the curve , between x = 0 and x= 2

3xy

2

0

3dxxA2

0

4

41

x

44 0241

e.g. (i) Find the area under the curve , between x = 0 and x= 2

3xy

2

0

3dxxA2

0

4

41

x

44 0241

2units4

e.g. (i) Find the area under the curve , between x = 0 and x= 2

3xy

2

0

3dxxA2

0

4

41

x

44 0241

3

2

2 1 dxxii

2units4

e.g. (i) Find the area under the curve , between x = 0 and x= 2

3xy

2

0

3dxxA2

0

4

41

x

44 0241

3

2

3

31

xx

3

2

2 1 dxxii

2units4

e.g. (i) Find the area under the curve , between x = 0 and x= 2

3xy

2

0

3dxxA2

0

4

41

x

44 0241

3

2

3

31

xx

3

2

2 1 dxxii

22

3133

31 33

2units4

e.g. (i) Find the area under the curve , between x = 0 and x= 2

3xy

2

0

3dxxA2

0

4

41

x

44 0241

3

2

3

31

xx

3

2

2 1 dxxii

22

3133

31 33

2units4

322

5

4

3 dxxiii

5

4

2

21

x

5

4

3 dxxiii

5

4

2

21

x

8009

41

51

21

22

5

4

3 dxxiii

5

4

2

21

x

8009

41

51

21

22

Exercise 11A; 1

Exercise 11B; 1 aefhi, 2ab (i,ii), 3ace, 4b, 5a, 7*

5

4

3 dxxiii