Quiz Chapter 07.05: Gauss Quadrature Rule

1. \, \displaystyle\int_{5}^{10} f(x)dx is exactly

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

3. The value of\, \displaystyle\int_{0.2}^{2.2} xe^{x} \, dx by using the two-point Gauss quadrature rule is most nearly

4. 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\, \displaystyle\int_{a}^{b} f(x)dx is

5. A scientist develops an approximate formula for integration as

• • • \, \displaystyle\int_{a}^{b} f(x)dx \approx c_{1}f(x_{1}),

where a \leq x_{1} \leq b

The values of c_{1} and x_{1} are found by assuming that the formula is exact for the functions of the form a_{0}x + a_{1}x^{2} polynomial. Then the resulting formula would therefore be exact for integrating

6. 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, bu you do not know how it varies. The flow rate Q is given by

• • • Q = \displaystyle\int_{0}^{2} 2 \pi r V \, dr

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 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