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 3
Area  9

Integration
Area Under Curve
Area  11  12 
3 3
Area  9

Integration
Area Under Curve
10   11  Area  11  12 
3 3 3 3
Area  9

Integration
Area Under Curve
10   11  Area  11  12 
3 3 3 3
1  Area  9

Integration
Area Under Curve
10   11  Area  11  12 
3 3 3 3
1  Area  9
Estimate Area  5 unit 2

Integration
Area Under Curve
10   11  Area  11  12 
3 3 3 3
1  Area  9
Estimate Area  5 unit 2
Exact Area  4 unit 2

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 unit 2

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  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  1.43  1.63  1.83  23 
3.24  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  1.43  1.63  1.83  23 
3.24  Area  4.84
Estimate Area  4.04 unit 2

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 x
A(c) is the area from 0 to c

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

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  x 

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(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

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.

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
h 0
 as h  0, c  x 
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
h 0
 as h  0, c  x 
h
 A 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
h 0
 as h  0, c  x 
h
 A x 
 the equation of the curve is the derivative of the Area function.

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
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;

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
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

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
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 

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
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 

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
  x4 
1
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
  x4 
1
4 0
 
 2  04 
1 4
4

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
  x4 
1
4 0
 
 2  04 
1 4
4
 4 units 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
  x4 
1
4 0
 
 2  04 
1 4
4
 4 units 2
3
ii   x 2  1dx
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
  x4 
1
4 0
 
 2  04 
1 4
4
 4 units 2
3 3
ii   x  1dx  
2 1 x 3  x 
2
2 3 

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
  x4 
1
4 0
 
 2  04 
1 4
4
 4 units 2
3 3
ii   x  1dx  
2 1 x 3  x 
2
2 3 
1 33  3  1 2 3  2
  
 3   3 

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
  x4 
1
4 0
 
 2  04 
1 4
4
 4 units 2
3 3
ii   x  1dx  
2 1 x 3  x 
2
2 3 
1 33  3  1 2 3  2
  
 3   3 
22

3

5
iii   x 3dx
4

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

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

5 5
 1  2 
iii   x dx   x 
3
4  2 4
   2  2
1 1 1
 
2 5 4 
9

800
Exercise 11A; 1
Exercise 11B; 1 aefhi, 2ab (i,ii), 3ace, 4b, 5a, 7*