is exactly

Two-point Gauss Quadrature Rule will give the same results as

1-segment Trapezoidal rule

Simpson’s 1/3^rd Rule

Exact integral

2-segment Trapezoidal rule

The value of  by using two-point Gauss Quadrature Rule is most nearly

11.672

11.807

12.811

14.633

A scientist uses one-point Gauss Quadrature Rule based on getting exact results of integration for functions  and .  The one-point rule approximation for

A scientist develops an approximate formula for integration as

where

The values of c₁ and x₁ are found by assuming that the formula is exact for the functions of the form a₀x + a₁x² polynomial.  Then the resulting formula would be exact for integrating

You are asked to estimate the water flow rate in a pipe of radius 2m at a remote area location with a harsh environment.  You already know that velocity varies along the radial location, but do not know how it varies.  The flow rate, Q is given by

To save money, you are allowed to put only two velocity probes (these probes send the data to the central office in New York, NY via satellite) in the pipe.  Radial location, r is measured from the center of the pipe, that is r=0 is the center of the pipe and r=2m is the pipe radius.   The radial locations you would suggest for the two velocity probes for the most accurate calculation of the flow rate are 

0,2

1,2

0,1

0.42,1.58

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