and
is
given by
so
that the width of the strip is multiplied by the
–
value at the midpoint.
A Level Maths Notes: C3 – The Mid Ordinate Rule
Also called the midpoint rule – the mid ordinate rule is another method to numerically estimate integrals. It states:
If an area of integration is
divided into n strips, the area of the strip betweenandis
given byso
that the width of the strip is multiplied by the–
value at the midpoint.
We do this for all n strips
obtaining
If
the strips are all of the same width
where
and
are
the limits of integration, then we can write,
Example: Using the mid
ordinate rule with 5 strips, estimate the vale of the integral
to
three decimal places. Evaluate the integral and compare the
approximation given by the ordinate rule with the true value.
Complete
the table of values for
Since
we must estimate the value of the integral to three decimal places,
we calculate values to four decimal places. The final answer is
quoted to three decimal places.
|
I |
0 |
1 |
2 |
3 |
4 |
5 |
|
|
0 |
0.2 |
0.4 |
0.6 |
0.8 |
1 |
|
|
0.1 |
0.3 |
0.5 |
0.7 |
0.9 |
|
|
|
1.1052 |
1.3499 |
1.6487 |
2.0138 |
2.4596 |
|
The % error is
