Numberical Methods, and Approximations

[scott73]

Hello again,
I am having trouble starting this problem, I am given the formula

INTEGRAL (from 0 to .5) of (sin(sqrt(x))), and asked to figure out the answer by approximation to the 5th decimal place.

I know that once I have an appropriatte N value, I can figure out the problem with the help of the trapezoidal rule. The problem I am having is figuring out N. I cannot seem to understand how you use the error test to figure out the minimum N that you need to use in order to get a problem accurate to the 5th decimal place. Any help would be great
Thanks

 

[pka]

The answer given by MathCad is 0.224126256944.
The problem with using trapezoidal rule is that the second derivative of \(\displaystyle \sin \left( {\sqrt x } \right)\) is not bounded on [0, 0.5].

Can you is an infinite series? \(\displaystyle \sin (x) = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k x^{2k + 1} }}{{\left( {2k + 1} \right)!}}}\)

Integrate this term by term:
\(\displaystyle \sin (\sqrt x ) = \sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k x^{\frac{{2k + 1}}{2}} }}{{\left( {2k + 1} \right)!}}}\)

 

[scott73]

Yes that worked, thank you very much, I was mistaken by thinking you had to use the trapezoidal rule because much of my problems around this one were using that rule. Thank you again