Approximating Definite Integrals — Question 10

Question 10

For Simpson’s Rule on $[0,\pi]$, suppose $$|f^{(4)}(x)|\le1.$$ Find the smallest even $n$ that guarantees error at most $10^{-6}$.

Question 10 – Solution

Step 1: Write Simpson’s error bound. $$|E_S|\le\frac{K(b-a)^5}{180n^4}.$$ Here $K=1$ and $b-a=\pi$, so $$|E_S|\le\frac{\pi^5}{180n^4}.$$ Step 2: Require the bound to be at most $10^{-6}$. $$\begin{align*} \frac{\pi^5}{180n^4}&\le10^{-6},\\ n^4&\ge\frac{\pi^5}{180\cdot10^{-6}},\\ n&\ge\left(\frac{\pi^5}{180\cdot10^{-6}}\right)^{1/4}\\ &\approx36.109. \end{align*}$$ Step 3: Enforce Simpson’s even-$n$ requirement. The first even integer at least $36.109$ is $38$. $$\boxed{n=38}$$