The Trapezium Rule
To find an area bounded by a curve, we need to
evaluate a definite integral.
If the integral cannot be evaluated, we can use an
approximate method.
This presentation uses the approximate method called
the Trapezium Rule.
The Trapezium Rule
The area under the curve is divided into a
number of strips of equal width. The top edge
of each strip . . .
. . . is replaced by a straight line so the
strips become trapezia.
The Trapezium Rule
The area under the curve is divided into a
number of strips of equal width. The top edge
of each strip . . .
. . . is replaced by a straight line so the
strips become trapezia.
The total area of the trapezia gives an
approximation to the area under the curve.
The formula for the area of a trapezium is:
the average of the parallel sides
 the distance apart
y0
y1
h
1
 ( y 0  y1 )  h
2
h seems a strange letter to use for width but it is
always used in the trapezium rule
The Trapezium Rule
1
e.g.1
0 1  x 2 dx
1
Suppose we have 5 strips.
y
1
1 x2
The Trapezium Rule
1
e.g.1
0 1  x 2 dx
1
y
Suppose we have 5 strips.
The parallel sides of
the trapezia are the
y-values of the
function.
The area of the
1st
y0
y1
y2
1
1 x2
y3 y
4
1
trapezium  ( y 0  y1 )  0  2
2
y5
The Trapezium Rule
1
e.g.1
0 1  x 2 dx
1
y
Suppose we have 5 strips.
The parallel sides of
the trapezia are the
y-values of the
function.
The area of the
1st
The area of the 2nd
y0
y1
y2
1
1 x2
y3 y
4
y5
1
trapezium  ( y 0  y1 )  0  2
2
1
trapezium  ( y1  y 2 )  0  2
2
etc.
The Trapezium Rule
1
0 1  x 2 dx
e.g.1
1
Suppose we have 5 strips.
Adding the areas of the
trapezia, we get
y
y0
y1
y2
1
1 x2
y3 y
4
y5
1
1
1
( y 0  y1 )  0  2  ( y1  y 2 )  0  2  . . .  ( y 4  y 5 )  0  2
2
2
2
1
  0  2 ( y 0  y1  y1  y 2  y 2  . . .  y 5 )
2
( removing common factors ).
Since y 1 is the side of 2 trapezia, it occurs twice
in the formula. The same is true for y 2 to y 4 .
1
1
1
dx
  0  2 ( y 0  2( y1  y 2  y 3  y 4 )  y 5 )
So,
2
2
1 x
0
The Trapezium Rule
1
0 1  x 2 dx
e.g.1
1
y
For each value of x, we
calculate the y-values, using
the function, and writing the
values in a table.
y0
y1
y2
1
1 x2
y3 y
4
y5
x
0
0 2
0 4
0 6
0 8
1 0
y
1
0  9615
0  8621
0  7353
0  6098
0 5
1
So,
0
1
dx   0  2 ( y 0  2( y1  y 2  y 3  y 4 )  y 5 )
2
1 x2
1
 0  1 (1  2 ( 0  9615  0  8621 0  7353  0  6098)  0  5 )
 0 784 ( 3.s.f. )
The Trapezium Rule
The general formula for the trapezium rule is
b

a
h
y dx 
( y 0  2( y1  y 2  y 3  . . .  y n1 )  y n )
2
where n is the number of strips.
The y-values are called ordinates.
There is always 1 more ordinate than the number
of strips.
The width, h, of each strip is given by
ba
h 
n
To improve the accuracy we just need to use
more strips.
The Trapezium Rule
e.g.2 Find the approximate value of


sin x dx
0
using 6 strips and giving the answer to 2 d. p.
Solution:

b
a
h
y dx 
( y 0  2( y1  y 2  y 3  y 4  y 5 )  y 6 )
2
Always draw a sketch showing the correct
b  number
a
of strips, even if the shape is wrong,
h  as it makes
n
it easy to find h and avoids errors.
h
 0 
 h 

6
6
The top edge of every trapezium in this
st
(
It
doesn’t
matter
that
the
1
example lies below the curve, so the
and last “trapezia”
are triangles )
rule will underestimate
the answer.

x
y

0
0
The Trapezium Rule
h
sin x dx 
( y 0  2( y1  y 2  y 3  y 4  y 5 )  y 6 )
2
Radians!

2
6
6
h 

4
6
5
6
0
0  8660 1
0  8660 0  5
0
To make sure that we have the required accuracy
0
0 5
3
6
 0  5236
we must use at least 1 more d. p. than the answer
6 still to store the values in the
2
requires.
It is better
 calculator’s memories.
sin x dx 
0
0  5236
(0  2 ( 0  5  0  8660  1  0  8660  0  5 )  0 )
2
 1 95 ( 2 d. p. )
The Trapezium Rule
The exact value of


sin x dx
is 2.
0
The percentage error in our answer is found as
follows:
error
Percentage error =
 100
exact value
( where, error = exact value  the approximate value ).
2  1  95

 100
2
 2 5 %
The Trapezium Rule
SUMMARY
 The trapezium law for estimating an area is

b
a
h
y dx 
( y 0  2( y1  y 2  y 3  . . .  y n1 )  y n )
2
where n is the number of strips.
ba
 The width, h, of each strip is given by h 
n
( but should be checked on a sketch )
 The number of ordinates is 1 more than the number
of strips.
 The trapezium law underestimates the area if
the tops of the trapezia lie under the curve and
overestimates it if the tops lie above the curve.
 The accuracy can be improved by increasing n.
The Trapezium Rule
Exercises
Use the trapezium rule to estimate the areas given
by the integrals, giving the answers to 3 s. f.
1.

1
(1  x ) dx using 4 strips
0
How can your answer be improved?
2.


2
0
cos x dx
using 4 ordinates
Find the approximate percentage error in your
answer, given that the exact value is 1.
The Trapezium Rule
Solutions
1.

1
0
n  4,
h  0 25
y  1 x
(1  x ) dx
using 4 strips
x
0
0  25
0 5
0  75
y
1
1  118
1  225 1  323
1
1  414
h
A
( y 0  2( y1  y 2  y 3 )  y 4 )
2
 1 22 ( 3 s. f. )
The answer can be improved by using more strips.
The Trapezium Rule
Solutions
2.

y  cos x

2
cos x dx
0
using 4 ordinates
n  3,

h
6
x
0
y
1

6
2
6
0  8660 0  5

2
0
h
A
( y 0  2( y1  y 2 )  y 3 )  0  977 ( 3 s. f. )
2
The exact value is 1, so the percentage error
1  0  977

 100
1
 2 3 %