Q1.  is exactly

Q2. For a definite integral of any third order polynomial, the two-point Gauss quadrature rule will give the same results as the

1-segment trapezoidal rule

2-segment trapezoidal rule

3-segment trapezoidal rule

Simpson's 1/3 rule

Q3. The value of  by using the two-point Gauss quadrature rule is most nearly

11.672

11.807

12.811

14.633

Q4. A scientist uses the one-point Gauss quadrature rule based on getting exact results of integration for functions f(x)=1 and x.  The one-point Gauss quadrature rule approximation for

Q5. 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 therefore be exact for integrating

Q6. 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 you 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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