Integration
Area Under Curve
Integration
Area Under Curve
Integration
Area Under Curve
Integration
Area Under Curve
Area  11  12 
3

3
Integration
Area Under Curve
Area  11  12 
3

Area  9

3
Integration
Area Under Curve
Area  11  12 
3

Area  9

3
Integration
Area Under Curve
10   11  Area  11  12 
3

3

3

Area  9

3
Integration
Area Under Curve
10   11  Area  11  12 
3

3

1  Area  9

3

3
Integration
Area Under Curve
10   11  Area  11  12 
3

3

3

1  Area  9
Estimate Area  5 unit 2

3
Integration
Area Under Curve
10   11  Area  11  12 
3

3

3

1  Area  9
Estimate Area  5 unit 2

Exact Are...
Area  0.40.43  0.83  1.23  1.63  23 
Area  0.40.43  0.83  1.23  1.63  23 
Area  5.76
Area  0.40.43  0.83  1.23  1.63  23 
Area  5.76
0.403  0.43  0.83  1.23  1.63   Area  0.40.43  0.83  1.23  1.63  23 
Area  5.76
0.403  0.43  0.83  1.23  1.63   Area  0.40.43  0.83  1.23  1.63  23 
2.56  Area  5.76
0.403  0.43  0.83  1.23  1.63   Area  0.40.43  0.83  1.23  1.63  23 
2.56  Area  5.76
Estimate Area  4.16...
Area  0.20.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  23 
Area  0.20.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  23 
Area  4.84
Area  0.20.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  23 
Area  4.84
0.203  0.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  

Area  0.20.23  0.43  0.63  0.83  13  1.23 ...
0.203  0.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  

Area  0.20.23  0.43  0.63  0.83  13  1.23 ...
0.203  0.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  

Area  0.20.23  0.43  0.63  0.83  13  1.23 ...
As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)

x
As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)

x
As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)

c
A(c) i...
As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)

c

x

A(...
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
A x   Ac    x  c  f ...
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    x  c  f ...
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    x  c  f ...
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    x  c  f ...
i.e. as h  0, the Area becomes exact
i.e. as h  0, the Area becomes exact
Ac  h   Ac 
f  x   lim
h 0
h
i.e. as h  0, the Area becomes exact
Ac  h   Ac 
f  x   lim
h 0
h
A x  h   A x 
 lim
 as h  0, c  x ...
i.e. as h  0, the Area becomes exact
Ac  h   Ac 
f  x   lim
h 0
h
A x  h   A x 
 lim
 as h  0, c  x ...
i.e. as h  0, the Area becomes exact
Ac  h   Ac 
f  x   lim
h 0
h
A x  h   A x 
 lim
 as h  0, c  x ...
i.e. as h  0, the Area becomes exact
Ac  h   Ac 
f  x   lim
h 0
h
A x  h   A x 
 lim
 as h  0, c  x ...
i.e. as h  0, the Area becomes exact
Ac  h   Ac 
f  x   lim
h 0
h
A x  h   A x 
 lim
 as h  0, c  x ...
i.e. as h  0, the Area becomes exact
Ac  h   Ac 
f  x   lim
h 0
h
A x  h   A x 
 lim
 as h  0, c  x ...
i.e. as h  0, the Area becomes exact
Ac  h   Ac 
f  x   lim
h 0
h
A x  h   A x 
 lim
 as h  0, c  x ...
e.g. (i) Find the area under the curve y  x 3 , between x = 0 and
x= 2
e.g. (i) Find the area under the curve y  x 3 , between x = 0 and
2
x= 2
A   x 3 dx
0
e.g. (i) Find the area under the curve y  x 3 , between x = 0 and
2
x= 2
A   x 3 dx
0

2

1 x 4 

4 0

e.g. (i) Find the area under the curve y  x 3 , between x = 0 and
2
x= 2
A   x 3 dx
0

2

1 x 4 

4 0



1 4
2...
e.g. (i) Find the area under the curve y  x 3 , between x = 0 and
2
x= 2
A   x 3 dx
0

2

1 x 4 

4 0



1 4
2...
e.g. (i) Find the area under the curve y  x 3 , between x = 0 and
2
x= 2
A   x 3 dx
0

2

1 x 4 

4 0



1 4
2...
e.g. (i) Find the area under the curve y  x 3 , between x = 0 and
2
x= 2
A   x 3 dx
0

2

1 x 4 

4 0



1 4
2...
e.g. (i) Find the area under the curve y  x 3 , between x = 0 and
2
x= 2
A   x 3 dx
0

2

1 x 4 

4 0



1 4
2...
e.g. (i) Find the area under the curve y  x 3 , between x = 0 and
2
x= 2
A   x 3 dx
0

2

1 x 4 

4 0



1 4
2...
5

iii   x 3dx
4
5

5

 1  2 
3
iii   x dx   x 
 2 4
4
5

5

 1  2 
3
iii   x dx   x 
 2 4
4
11 1 
   2  2
2 5 4 
9

800
5

5

 1  2 
3
iii   x dx   x 
 2 4
4
11 1 
   2  2
2 5 4 
9

800

Exercise 11A; 1
Exercise 11B; 1 ...
11 x1 t16 01 area under curve (2013)
11 x1 t16 01 area under curve (2013)
11 x1 t16 01 area under curve (2013)
11 x1 t16 01 area under curve (2013)
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11 x1 t16 01 area under curve (2013)

  1. 1. Integration Area Under Curve
  2. 2. Integration Area Under Curve
  3. 3. Integration Area Under Curve
  4. 4. Integration Area Under Curve Area  11  12  3 3
  5. 5. Integration Area Under Curve Area  11  12  3 Area  9 3
  6. 6. Integration Area Under Curve Area  11  12  3 Area  9 3
  7. 7. Integration Area Under Curve 10   11  Area  11  12  3 3 3 Area  9 3
  8. 8. Integration Area Under Curve 10   11  Area  11  12  3 3 1  Area  9 3 3
  9. 9. Integration Area Under Curve 10   11  Area  11  12  3 3 3 1  Area  9 Estimate Area  5 unit 2 3
  10. 10. Integration Area Under Curve 10   11  Area  11  12  3 3 3 1  Area  9 Estimate Area  5 unit 2 Exact Area  4 unit  2 3
  11. 11. Area  0.40.43  0.83  1.23  1.63  23 
  12. 12. Area  0.40.43  0.83  1.23  1.63  23  Area  5.76
  13. 13. Area  0.40.43  0.83  1.23  1.63  23  Area  5.76
  14. 14. 0.403  0.43  0.83  1.23  1.63   Area  0.40.43  0.83  1.23  1.63  23  Area  5.76
  15. 15. 0.403  0.43  0.83  1.23  1.63   Area  0.40.43  0.83  1.23  1.63  23  2.56  Area  5.76
  16. 16. 0.403  0.43  0.83  1.23  1.63   Area  0.40.43  0.83  1.23  1.63  23  2.56  Area  5.76 Estimate Area  4.16 unit 2
  17. 17. Area  0.20.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  23 
  18. 18. Area  0.20.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  23  Area  4.84
  19. 19. Area  0.20.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  23  Area  4.84
  20. 20. 0.203  0.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83   Area  0.20.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  23  Area  4.84
  21. 21. 0.203  0.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83   Area  0.20.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  23  3.24  Area  4.84
  22. 22. 0.203  0.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83   Area  0.20.23  0.43  0.63  0.83  13  1.23  1.43  1.63  1.83  23  3.24  Area  4.84 Estimate Area  4.04 unit 2
  23. 23. As the widths decrease, the estimate becomes more accurate, lets investigate one of these rectangles. y y = f(x) x
  24. 24. As the widths decrease, the estimate becomes more accurate, lets investigate one of these rectangles. y y = f(x) x
  25. 25. As the widths decrease, the estimate becomes more accurate, lets investigate one of these rectangles. y y = f(x) c A(c) is the area from 0 to c x
  26. 26. As the widths decrease, the estimate becomes more accurate, lets investigate one of these rectangles. y y = f(x) c x A(c) is the area from 0 to c A(x) is the area from 0 to x x
  27. 27. A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle;
  28. 28. A(x) – A(c) denotes the area from c to x, and can be estimated by the rectangle; f(x) x-c
  29. 29. 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    x  c  f  x 
  30. 30. 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    x  c  f  x  A x   Ac  f x  xc
  31. 31. 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    x  c  f  x  A x   Ac  f x  xc Ac  h   Ac   h h = width of rectangle
  32. 32. 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    x  c  f  x  A x   Ac  f x  xc Ac  h   Ac   h = width of rectangle h As the width of the rectangle decreases, the estimate becomes more accurate.
  33. 33. i.e. as h  0, the Area becomes exact
  34. 34. i.e. as h  0, the Area becomes exact Ac  h   Ac  f  x   lim h 0 h
  35. 35. i.e. as h  0, the Area becomes exact Ac  h   Ac  f  x   lim h 0 h A x  h   A x   lim  as h  0, c  x  h 0 h
  36. 36. i.e. as h  0, the Area becomes exact Ac  h   Ac  f  x   lim h 0 h A x  h   A x   lim  as h  0, c  x  h 0 h  A x 
  37. 37. i.e. as h  0, the Area becomes exact Ac  h   Ac  f  x   lim h 0 h A x  h   A x   lim  as h  0, c  x  h 0 h  A x   the equation of the curve is the derivative of the Area function.
  38. 38. i.e. as h  0, the Area becomes exact Ac  h   Ac  f  x   lim h 0 h A x  h   A x   lim  as h  0, c  x  h 0 h  A x   the equation of the curve is the derivative of the Area function. The area under the curve y  f  x  between x  a and x  b is;
  39. 39. i.e. as h  0, the Area becomes exact Ac  h   Ac  f  x   lim h 0 h A x  h   A x   lim  as h  0, c  x  h 0 h  A x   the equation of the curve is the derivative of the Area function. The area under the curve y  f  x  between x  a and x  b is; b A   f  x dx a
  40. 40. i.e. as h  0, the Area becomes exact Ac  h   Ac  f  x   lim h 0 h A x  h   A x   lim  as h  0, c  x  h 0 h  A x   the equation of the curve is the derivative of the Area function. The area under the curve y  f  x  between x  a and x  b is; b A   f  x dx a  F b   F a 
  41. 41. i.e. as h  0, the Area becomes exact Ac  h   Ac  f  x   lim h 0 h A x  h   A x   lim  as h  0, c  x  h 0 h  A x   the equation of the curve is the derivative of the Area function. The area under the curve y  f  x  between x  a and x  b is; b A   f  x dx a  F b   F a  where F  x  is the primitive function of f  x 
  42. 42. e.g. (i) Find the area under the curve y  x 3 , between x = 0 and x= 2
  43. 43. e.g. (i) Find the area under the curve y  x 3 , between x = 0 and 2 x= 2 A   x 3 dx 0
  44. 44. e.g. (i) Find the area under the curve y  x 3 , between x = 0 and 2 x= 2 A   x 3 dx 0 2 1 x 4   4 0 
  45. 45. e.g. (i) Find the area under the curve y  x 3 , between x = 0 and 2 x= 2 A   x 3 dx 0 2 1 x 4   4 0   1 4 2  04  4
  46. 46. e.g. (i) Find the area under the curve y  x 3 , between x = 0 and 2 x= 2 A   x 3 dx 0 2 1 x 4   4 0   1 4 2  04  4  4 units 2
  47. 47. e.g. (i) Find the area under the curve y  x 3 , between x = 0 and 2 x= 2 A   x 3 dx 0 2 1 x 4   4 0   1 4 2  04  4  4 units 2 3 ii   x 2  1dx 2
  48. 48. e.g. (i) Find the area under the curve y  x 3 , between x = 0 and 2 x= 2 A   x 3 dx 0 2 1 x 4   4 0   1 4 2  04  4  4 units 2 3 3 1 x 3  x  2 ii   x  1dx   2 3  2
  49. 49. e.g. (i) Find the area under the curve y  x 3 , between x = 0 and 2 x= 2 A   x 3 dx 0 2 1 x 4   4 0   1 4 2  04  4  4 units 2 3 3 1 x 3  x  2 ii   x  1dx   2 3  2 1 33  3  1 2 3  2     3 3    
  50. 50. e.g. (i) Find the area under the curve y  x 3 , between x = 0 and 2 x= 2 A   x 3 dx 0 2 1 x 4   4 0   1 4 2  04  4  4 units 2 3 3 1 x 3  x  2 ii   x  1dx   2 3  2 1 33  3  1 2 3  2     3 3     22  3
  51. 51. 5 iii   x 3dx 4
  52. 52. 5 5  1  2  3 iii   x dx   x   2 4 4
  53. 53. 5 5  1  2  3 iii   x dx   x   2 4 4 11 1     2  2 2 5 4  9  800
  54. 54. 5 5  1  2  3 iii   x dx   x   2 4 4 11 1     2  2 2 5 4  9  800 Exercise 11A; 1 Exercise 11B; 1 aefhi, 2ab (i,ii), 3ace, 4b, 5a, 7*
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