Numberical Methods, and Approximations

scott73

New member
Joined
Sep 27, 2005
Messages
8
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

Elite Member
Joined
Jan 29, 2005
Messages
10,323
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

New member
Joined
Sep 27, 2005
Messages
8
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
 
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