The transverse intensity profile of a laser beam is modelled by (arbitrary units

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Integration: error expansions, Gauss rules and adaptivity

Problem

The transverse intensity profile of a laser beam is modelled by

I(x)=e^{-x^2/2}

(arbitrary units,

x

in mm). The power collected by a detector spanning

0\le x\le2

mm is

E=\int_0^2 I(x)\,\mathrm{d}x

, whose exact value (from the standard normal cumulative distribution function) is

E=1.19629

(to six figures). ((a)) Using the standard two-point Gauss-Legendre rule on

[-1,1]

, with nodes

\pm1/\sqrt3

and weights

1,1

, transform the nodes to

[0,2]

. ((b)) Evaluate the resulting two-point Gauss estimate of

E

. ((c)) State the error against the exact value given above.

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