Approximating the value of an integral

[burt]

"To approximate the value of \(\int^{0.2}_0\frac{1}{1+x^4}\ dx\) up to 6 decimal places, how many terms are needed in its series representation?"

I was given this problem, and I'm not sure what to do with it. How do I ensure accuracy up to a number of decimal places for an integral?
(Sorry that I'm not showing any work - it's because I don't know where to start because I don't understand what that basic process is. I want to understand it - not just for this problem.)

 

[Romsek]

I suspect there's a formula out there that gives you the error bounds on integration using Taylor series directly. A couple minutes on google didn't find it but it's probably out there.

That said we can always just start integrating the Taylor series, each term is easily integrated, and stop when we've got 6 decimal places that aren't changing.

The Taylor series of [MATH]\dfrac{1}{1+x^4}[/MATH] about \(\displaystyle x_0=0\) is given by

[MATH]\dfrac{1}{1+x^4} = 1 - x^4 + x^8 -x^{12}+ \dots + (-x)^{4k},~k \in \mathbb{N^*}[/MATH]
This is simple enough to integrate term by term

[MATH]\displaystyle \int_0^{0.2} \dfrac{1}{1+x^4} \approx \left. x \right|_{0}^{0.2} - \left . \dfrac{x^5}{5}\right|_0^{0.2} + \dots[/MATH]
You can set this all up in excel or your software of choice and just keep adding in terms until you have your 6 stable digits. It doesn't take very many terms.