1. Evaluate the integral of the data that is tabulated below, with:
a) the trapezoidal rule
b) and Simpson's rules:
X | -2 | 0 | 2 | 4 | 6 | 8 | 10
f(x) | 35 | 5 | -10 | 2 | 5 | 3 | 20
2. The function f(x) = 2e^-1.5x can be used to generate the following table of unevenly spaced data:
X | 0 | 0.05 | 0.15 | 0.25 | 0.35 | 0.475 | 0.6
f(x) | 2 | 1.8555 | 1.5970 | 1.3746 | 1.1831 | 0.9808 | 0.8131
3. Use Romberg integration of order h^8 to evaluate:
∫[0 to 3] xe^x dx
Compare ε_a ε_r
Obtain an estimate of the integral of Problem 3, but use Gauss-Legendre formulas with two, three and four points. Calculate ε_t for each case based on the solution analytics.
4. Use numerical integration to evaluate the following:
a. ∫[2 to ∞] dx / x(x+2)
b. ∫[0 to ∞] e^-y sin^2 y dy
c. ∫[0 to ∞] 1 / ((1+y^2)(1+y^2/2)) dy
d. ∫[-2 to ∞] ye^-y dy
e. ∫[0 to ∞] (1/√2) e^(-x^2/2) dx