Thread: estimate the integral

Just got out of my Analysis final and one of the questions was:

Estimate $\displaystyle f(x) = \int_0^1 \sin(x^2)dx$ to within $\displaystyle 10^{-3}.$

I may have gotten it right, but does anyone have any ideas?

I would probably have expanded $\displaystyle \sin (x^2)$ with a Taylor series and then integrated term by term (since the convergence is uniform). Though how many terms I am not quite sure.

That's what I did, only I'm not sure if my taylor expansion was right.

Originally Posted by Anonymous1

That's what I did, only I'm not sure if my taylor expansion was right.

$\displaystyle
\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \pm \cdots
$ BTW - you only need two terms.