Approximation of $T10$ for integral $\int_0^1\sin(x^2) dx$ Trapezoid Approximation

Question

I got through most of the work with finding the approximation of $T10$ which comes out to be $=.3111708111$, I also found the error of $Et10$ when I plugged into the formula of $K(b-a)^3/12(n)^2$ . My question is how do i find the smallest value of $n$ so that the approximation of $Tn$ is accurate to within $0.00001$? Thank You

Answer 1

Since you have an upper bound for the error $E\leq\frac{K(b-a)^3}{12n^2}$, and you desire $E<10^{-5}$, you can ensure this is true by making $$ \frac{K(b-a)^3}{12n^2}<10^{-5}. $$

You now need to solve this inequality for $n$.

The solution is $\displaystyle n>\sqrt{\frac{K(b-a)^3}{12\times10^{-5}}}$ and you can substitute in the appropriate values for $K$, $a$ and $b$.