I have an integral:
$$\int_0^1sinx^2dx$$
Task is to solve this integral using Simpson's rule with precision $\frac{1}{2}10^{-4}$.
I am not sure how should I do that. I have this formula for calculating error:
$$r(\frac{h}{2}) = \frac{I(h) - I(\frac{h}{2})}{k^2 - 1}$$ where $$I = f(x)dx$$ in this case would be: $$I = sinx^2dx$$
(So $I=sinx^2dx$ is same as $I=2xcosx^2$)
Should this be correct way to solution:
$a=0$ and $b=1$, so first I should take some initial $k$ and divide $[a,b]$ into $k$ equal pieces where $h$ would be step. Then plug $h$ and $k$ into formula and see the result. If result is greater then $\frac{1}{2}10^{-4}$ then increase $k$. Repeat until next criteria is satisfied:
$$r(\frac{h}{2}) \le \frac{1}{2}10^{-4}$$.
And at last, I have $h$, now I can simply calculate integral using Simpson's rule and get desired precision.
Use the error bound $\frac{(b-a)^5}{90 \cdot 2^5} f^{(4)}(\xi)$ for the one panel Simpson's rule instead.