[Solved] MA 322 Lab 5

1. Approximate the following integrals using Gaussian quadrature with n = 2, and compare your results to the exact values of the integrals

dx dx dx

2. Let x₁,,x_n [a,b] , w₁,,w_n R be the nodes and the weight of a quadrature formula. Assume that w_j < 0 for some j 1,,n. Construct a continuous function f : [a,b] R such that f(x) 0, x [a,b], i.e.,

f(x)dx > 0,

but

.

3. Use the two-point Gaussian quadrature rule to approximate

dx

and compare the result with the trapezoidal and Simpsons rules. 4. Use the three-point Gaussian quadrature formula to evaluate

.

Compare this result with that obtained by Simpsons rule with h = 0.125. 5. There are two Newton-Cotes formulas for n = 2; namely,

,

,

Which is better?

6. Use the n = 1,2,3,4,5 point Gaussian quadrature formula to evaluate

dx

to 2 correct decimal places.

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