27. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
x
28. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
x
29. 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
30. As the widths decrease, the estimate becomes more accurate, lets
investigate one of these rectangles.
y
y = f(x)
c x x
A(c) is the area from 0 to c
A(x) is the area from 0 to x
31. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
32. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
33. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
34. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
A x Ac
f x
xc
35. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
A x Ac
f x
xc
Ac h Ac
h = width of rectangle
h
36. A(x) – A(c) denotes the area from c to x, and can be estimated by
the rectangle;
f(x)
x-c
A x Ac x c f x
A x Ac
f x
xc
Ac h Ac
h = width of rectangle
h
As the width of the rectangle decreases, the estimate becomes more
accurate.
38. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
39. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
40. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
41. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
h
A x
the equation of the curve is the derivative of the Area function.
42. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
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;
43. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
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
44. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
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
45. i.e. as h 0, the Area becomes exact
Ac h Ac
f x lim
h 0 h
A x h A x
lim
h 0
as h 0, c x
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
46. e.g. (i) Find the area under the curve y x 3 , between x = 0 and
x= 2
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
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
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
2 04
1 4
4
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
2 04
1 4
4
4 units 2
51. 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
2 04
1 4
4
4 units 2
3
ii x 2 1dx
2
52. 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
2 04
1 4
4
4 units 2
3 3
ii x 1dx
2 1 x 3 x
2 3 2
53. 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
2 04
1 4
4
4 units 2
3 3
ii x 1dx
2 1 x 3 x
2 3 2
1 33 3 1 2 3 2
3 3
54. 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
2 04
1 4
4
4 units 2
3 3
ii x 1dx
2 1 x 3 x
2 3 2
1 33 3 1 2 3 2
3 3
22
3