I just learned about Taylor polynomials, and I am trying to estimate $\int_{0}^{1/2}\sin(x^2)dx$ using the 3rd degree Taylor polynomial of $F(x)=\int_{0}^{x}\sin(t^2)dt$ at $0$. I get the following:
F'(x)=\sin(x^2),
F''(x)=2x\cos(x^2),
$F^{(3)}(x)=2\cos(x^2)-4x^2\sin(x^2)$.
And so the estimate is:
\frac{F(0)}{0!}\cdot 1+\frac{F'(0)}{1!}\cdot \frac{1}{2}+\frac{F''(0)}{2!}\cdot \frac{1}{4}+\frac{F^{(3)}(0)}{3!}\cdot \frac{1}{8} = 0.
However this seems very weird (shouldn't it be closer to the actual value than $0$?), is my estimation correct?