Error bound for $\cos(x^3)$ under Mn, Tn and Sn

Question

In this problem we will approximate the integral of $\cos(x^3)$ over the interval $[0, 2]$.
(a) Write an expression for MN, TN and SN with $N = 4$.
(b) For each of the approximations determine an N so that the error is guaranteed to be less than $10^{-4}.$ You will find it useful to know that on the interval $[0, 2], |\frac {d^4\cos(x^3)}{dx^4} | \le 850$.

Please help me with the question. Thanks a lot!

Answer 1

Hints:

For Simpson's rule, we write:

$$\displaystyle \int_0^2 \cos x^3 dx = \frac{1}{12} \left(1 + 2 \sum_{n=1}^{4-1} \cos n^3 + 4 \sum_{n=1}^{4} \cos (\frac{1}{8} (-1+2 n)^3)+ \cos 8\right) = 0.878445$$

Can you now follow the approach and derive the trapezoidal and midpoint rule?

The results should be:

• Midpoint Rule = $1.07095$
• Trapezoidal Rule = $0.493433$
• Exact result = $0.8554746507540773$