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 between
and
is given by
so 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