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,...,xn ∈ [a,b] , w1,...,wn ∈R be the nodes and the weight of a quadrature formula. Assume that wj < 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 Simpson’s rules. 4. Use the three-point Gaussian quadrature formula to evaluate
.
Compare this result with that obtained by Simpson’s 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.
