Integration

1
Integration

• Integrationis the total value, or summation,
  of f(x) dx over the range from a to b

2
Composite Trapezoidal Rule

• Assuming n1 data points are evenly spaced, there
  will be n intervals over which to integrate.
• The total integral can be calculated by
  integrating each subinterval and then adding them
  together

3
Simpsons Rules

• More complicated approximation formulas can
  improve the accuracy for curves - these include
  using (a) 2nd and (b) 3rd order polynomials.
• The formulas that result from taking the
  integrals under these polynomials are called
  Simpsons rules.

4
Simpsons Rule

• Simpsons rule corresponds to using second-order
  polynomials
• Integration over the three points simplifies to

5
Composite Simpsons Rule

• Simpsons rule can be used on a set of
  subintervals in much the same way the trapezoidal
  rule was, except there must be an odd number of
  points.
• Because of the heavy weighting of the internal
  points, the formula is a little more complicated
  than for the trapezoidal rule

6
Simpsons 3/8 Rule

• Simpsons 3/8 rule corresponds to using
  third-order polynomials to fit four points.
  Integration over the four points simplifies
  to
• Simpsons 3/8 rule is generally used in concert
  with Simpsons 1/3 rule when the number of
  segments is odd.

7
Integration with Unequal Segments

• Previous formulas were simplified based on
  equispaced data points - though this is not
  always the case.
• The trapezoidal rule may be used with data
  containing unequal segments

8
MATLAB Functions

• MATLAB has built-in functions to evaluate
  integrals based on the trapezoidal rule
• z trapz(y)z trapz(x, y)produces the
  integral of y with respect to x. If x is omitted,
  the program assumes h1.
• z cumtrapz(y)z cumtrapz(x, y)produces the
  cumulative integral of y with respect to x. If x
  is omitted, the program assumes h1.

9
Example

• Approximate the distance travelled from the
  following experimental data
• Note, distance travelled is area under curve of
  velocity, ie integral

t 0 1 1.4 2 3 4.3 6 6.7 8
v 0 10 13 19 26 34 41 43 46
10
Gauss Quadrature

• Gauss quadrature describes a class of techniques
  for evaluating the area under a straight line by
  joining any two points on a curve rather than
  simply choosing the endpoints.
• The key is to choose the line that balances the
  positive and negative errors.

11
Gauss-Legendre Formulas

• The Gauss-Legendre formulas seem to optimize
  estimates to integrals for functions over
  intervals from -1 to 1.
• Integrals over other intervals require a change
  in variables to set the limits from -1 to 1.
• The integral estimates are of the formwhere
  the ci and xi are calculated to ensure that the
  method exactly integrates up to (2n-1)th order
  polynomials over the interval from -1 to 1.

12
Adaptive Quadrature

• Methods such as Simpsons 1/3 rule has a
  disadvantage in that it uses equally spaced
  points - if a function has regions of abrupt
  changes, small steps must be used over the entire
  domain to achieve a certain accuracy.
• Adaptive quadrature methods for integrating
  functions automatically adjust the step size so
  that small steps are taken in regions of sharp
  variations and larger steps are taken where the
  function changes gradually.

13
Adaptive Quadrature in MATLAB

• MATLAB has two built-in functions for
  implementing adaptive quadrature
• quad uses adaptive Simpson quadrature
• q quad(fun, a, b, tol, trace, p1, p2, )
• fun function to be integrated
• a, b integration bounds
• tol desired absolute tolerance (default 10-6)
• trace flag to display details or not
• p1, p2, extra parameters for fun
• quadl has the same arguments