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 🙂
Best Answer
To find Upper Bound of Error using Trapezoidal Rule
No. of sub intervals = $n$
Given integral is $$\int_0^\pi \sin(2x)\,\mathrm {d}x$$
$$\implies f(x)=\sin(2x), a=0,b=\pi$$
$$f'(x) = 2\cos(2x)$$
$$f''(x)=-4\sin(2x)$$
The maximum value of $|f''(x)|$ will be 4
$M=4$
The upper bound of error,
$$|e_T|\le \frac{M(b-a)^3}{12n^2}$$
$$|e_T|\le \frac{\pi^3}{3n^2}$$