Obtain an upper bound on the absolute error when we compute $\displaystyle\int_0^6 \sin x^2 \,\mathrm dx$ by means of the composite trapezoid rule using 101 equally spaced points.

The formula I'm trying to use is:

$$ I = \frac{h}{2} \sum_{i=1}^n \Big[f(x_{i-1}) + f(x_i)\Big] - \frac{h^3}{12} \sum_{i=1}^n f^{''}(\xi_i) $$

But I'm lost on how to calculate the error and find a value for $\xi$. What's the general way of finding the error like this? Thanks for any help :)


To find Upper Bound of Error using Trapezoidal Rule

No. of sub intervals = $n$

Given integral is $$\int_0^\pi \sin(2x)\,\mathrm {d}x$$

$$\implies f(x)=\sin(2x), a=0,b=\pi$$

$$f'(x) = 2\cos(2x)$$


The maximum value of $|f''(x)|$ will be 4


The upper bound of error,

$$|e_T|\le \frac{M(b-a)^3}{12n^2}$$

$$|e_T|\le \frac{\pi^3}{3n^2}$$

  • $\begingroup$ This isn't the same integral. Was he just giving a example? Or are you supposed to change the integral from $x^2$ to $2x$ in the problem for some reason... $\endgroup$
    – Shammy
    Nov 3 '15 at 2:55

You are not supposed to compute the exact values of the $\xi_i$. Instead, what can you say about $f''$? Can you find a simple expression $g(x)$ with $|f''(\xi)|\le g(\xi)$? And then find a simple estimate for the sum, given that each $\xi_i$ is in a known interval?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.