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Calculating the area under a curve given by a general set of points

An approach for calculating the integral of a function from uncertain, arbitrary measurements is applied to the Earth's radiation budget.

Determining the definite integral I of a function f over an interval is a basic concept in mathematics. For cases that cannot be treated analytically, quadrature rules are applied to obtain I numerically, ideally to within a user-prescribed accuracy. Typically, f is evaluated at stipulated points in order to implement the rule.

What can be done
  1. if f is given by measurement at a number of points,
  2. if these points are arbitrarily spaced,
  3. about the measurement uncertainties?
An approach is described based on
  1. regarding the measurement points as representing a curve, the area under which is sought,
  2. representing each interval between adjacent points by a polynomial piece,
  3. forming the definite integral of each such piece,
  4. summing these integrals to approximate the required area,
  5. comparing the results obtained using polynomials of different orders,
  6. propagating the measurement uncertainties to evaluate the uncertainties associated with the approximations.

An application to climate change is presented, viz., quantifying the component of the Earth's Radiation Budget relating to the incoming radiation from the sun.


Professor Maurice Cox | talks | www


Date and Time:

26 October 2005 at 1:30 pm


1 hour



School of Maths, Kingston University
School of Maths, Kingston University
Penrhyn Road
Kingston upon Thames
+44 20 8547 7922

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