Interpolation And Ploynomial Approximation

a) Suppose that the interval $[a, b]$ in $\int_{a}^{b} f(x) \, dx$ is divided into $N$ even intervals. Show that

$$
\int_{a}^{b} f(x) \, dx = h \left( \frac{1}{3} f_{j+2} + \frac{4}{3} f_{j+1} - \frac{1}{3} f_{j} \right) = \frac{h}{3} (f_{j+2} + 4f_{j+1} + f_{j})
$$

(5 marks)

b) Estimate $I = \int_{0}^{1} e^{-x^2} \, dx$ accurate to 2 d.p using trapezoidal rule and with $h = 0.1$.
(7 marks)
Find the exact solution to 4d.p, hence determine the relative error in the trapezoidal approximation.

c) Find the step size $h$ so that the absolute value of the error by Simpson's rule is less than $5 \times 10^{-9}$ when solving $\int_{2}^{4} \frac{dx}{2}$
(5 marks)

d) Determine the volume of a solid generated by revolution where radius $r(x)$, the perpendicular distance from the x-axis is

Using Simpson's rule with $n = 3$ and $h = 1$
(3 marks)