I'd like to ask if someone can please give me a little push with this assignment:
Approximate the value of the integral $\int_0^1 \sin(x^2) dx$ using only $\mathbb{N}$ numbers and basic operations $(+,-,*,/)$. For example use Simpson's rule or the Trapezoidal rule to calculate the integral, then use Taylor's series to determine the value of sinus.
I have used Simpson's rule to approximate the integral and got:
$\int_0^1 \sin(x^2) dx \approx \frac{2}{3}\sin(\frac{1}{4}) + \frac{1}{6}\sin(1)$
But I don't know what to do with the Taylor series. Should I compute it for $\frac{2}{3}\sin(x^2) + \frac{1}{6}\sin(x^2)$ and then express it for $x = \frac{1}{4}$ and $x = 1$ or is my thinking bad?