1. Numerical Integration:
Approximating an integral by a sum

2. Integrals as areas

3. Integrals as areas Approximate the integral by a finite sum of areas
rectangles

4. Integrals as areas Approximate the integral by a finite sum of areas
trapeziums

5. Integrals as areas
Trapezium rule:
Associated error:

6. Integrals as areas
Composite trapezium rule:

7. Integrals as areas
Simpson’s rule:

8. Integrals as areas
the rectangle approximation takes the function to be
constant in the interval
the trapezium rule uses linear interpolation between points
Simpson’s rule uses polynomial (quadratic) interpolation
between the points and has an associated error
these should be compared to a Taylor expansion, the first
term is a constant, the second term is linear, the third is
quadratic…

9. Simpson’s rule: derivation by method of undetermined
coefficients
express the integral as
this must hold for polynomials of degree two or less.
In particular, it must hold for
but we already know the left hand side for these

10. Simpson’s rule: derivation by method of undetermined
coefficients
Now calculate the right-hand-side:
Solving these gives

11. Romberg integration: integration by iteration
make repeated use of the trapezium rule
etc.
this is equivalent to
etc.
this gives a series of approximations which can be used to
extrapolate to give the answer

12. Romberg integration: doing the extrapolation
the algorithm takes the form of a triangle, Rjk, where we start with R00 and work down
R00
R10 R11
R20 R21 R22
R30 R31 R32 R33 … …
are the approximations, repeat until

13. Romberg integration: doing the extrapolation
The left hand column are given by the trapezium rule
starting with
To work from the left to the right for a given column use

14. so
Example,

15. We end up with the following triangle
Example,
We end up with the following triangle
First approximation
Second approximation
Third approximation

16. Infinite ranged integrals:
evaluate an integral over an infinite range
the previous methods would lead to an infinite sum
so cannot be used
transform the integral into a finite ranged integral, e.g.
change the variable in the second integral
(prove this)