Download - 11X1 T14 01 area under curves
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IntegrationArea Under Curve
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IntegrationArea Under Curve
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IntegrationArea Under Curve
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IntegrationArea Under Curve
33 2111Area
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IntegrationArea Under Curve
33 2111Area
9Area
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IntegrationArea Under Curve
33 2111Area
9Area
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IntegrationArea Under Curve
33 2111Area
9Area
33 1101
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IntegrationArea Under Curve
33 2111Area
9Area
33 1101
1
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IntegrationArea Under Curve
33 2111Area
9Area
33 1101
1
2unit5AreaEstimate
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IntegrationArea Under Curve
33 2111Area
9Area
33 1101
1
2unit5AreaEstimate
24 unitExact Area
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33333 26.12.18.04.04.0Area
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33333 26.12.18.04.04.0Area
76.5Area
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33333 26.12.18.04.04.0Area
76.5Area
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33333 26.12.18.04.04.0Area
76.5Area 33333 6.12.18.04.004.0
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33333 26.12.18.04.04.0Area
76.5Area 33333 6.12.18.04.004.0
56.2
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33333 26.12.18.04.04.0Area
76.5Area 33333 6.12.18.04.004.0
56.22unit4.16AreaEstimate
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3333333333 28.16.14.12.118.06.04.02.02.0Area
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3333333333 28.16.14.12.118.06.04.02.02.0Area 84.4Area
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3333333333 28.16.14.12.118.06.04.02.02.0Area 84.4Area
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3333333333 28.16.14.12.118.06.04.02.02.0Area 84.4Area
3333333333 8.16.14.12.118.06.04.02.002.0
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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
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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
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As the widths decrease, the estimate becomes more accurate, lets investigate one of these rectangles.
y
x
y = f(x)
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As the widths decrease, the estimate becomes more accurate, lets investigate one of these rectangles.
y
x
y = f(x)
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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
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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
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A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle;
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A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle;
f(x)
x - c
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A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle;
f(x)
x - c
xfcxcAxA
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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
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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
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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.
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exact becomesAreathe,0 as i.e. h
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exact becomesAreathe,0 as i.e. h
h
cAhcAxfh
0lim
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exact becomesAreathe,0 as i.e. h
h
cAhcAxfh
0lim
h
xAhxAh
0lim xch ,0as
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exact becomesAreathe,0 as i.e. h
h
cAhcAxfh
0lim
h
xAhxAh
0lim xch ,0as
xA
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exact becomesAreathe,0 as i.e. h
h
cAhcAxfh
0lim
h
xAhxAh
0lim xch ,0as
xA
function.Areatheofderivativetheiscurvetheofequation the
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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
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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
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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
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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
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e.g. (i) Find the area under the curve , between x = 0 and x= 2
3xy
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e.g. (i) Find the area under the curve , between x = 0 and x= 2
3xy
2
0
3dxxA
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e.g. (i) Find the area under the curve , between x = 0 and x= 2
3xy
2
0
3dxxA2
0
4
41
x
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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
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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
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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
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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
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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
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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
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5
4
3 dxxiii
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5
4
2
21
x
5
4
3 dxxiii
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5
4
2
21
x
8009
41
51
21
22
5
4
3 dxxiii
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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