Integration Area Under Curve
11 x1 t16 01 area under curve (2012)
Jun 23, 2015
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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
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