Description
Problem 1. Find a formula of the form
Z 1
−1
f(x)dx ≈ c
X
2
i=0
f(xi)
that is exact for all quadratic polynomials.
Problem 2. Show how the Gaussian quadrature rule
Z 1
−1
f(x)dx ≈
5
9
f(−
s
3
5
) + 8
9
f(0) + 5
9
f(
s
3
5
)
can be used for R a
b
f(x)dx. Apply this result to evaluate
(a) Z π/2
0
xdx
(b) Z 4
0
(sin t)/tdt.
Problem 3. Apply the Romberg algorithm to find R(2, 2) for these integrals:
(a)
Z 3
1
1/xdx
(b)
Z π/2
0
(x/π)
2
dx.
1
