APPROXIMATE THE INTEGRAL

[Ryan Rigdon]

I have this problem that is driving me up the wall. Every time i do it i get it wrong. the only thing i get write on it is the fact that n = 16

its the second part of the problem that i get wrong. is approximating the integral. i have followed examples and nothing. i get the answer of 1.253 and we were to round off to the 3rd decimal place. am i missing something. Why do i keep getting it wrong.

 

[Ryan Rigdon]

i see the problem i only went to X8 = 3.5 i need to keep going until i get to 6.

 

[Ryan Rigdon]

got it right

 

[BigGlenntheHeavy]

\(\displaystyle \int_{1}^{6}\frac{1}{x}dx \ \dot= \ 1.79175946923 \ (TI-89)\)

\(\displaystyle Now, \ using \ Simpson's \ Rule \ with \ n \ = \ 16, \ we \ get;\)

\(\displaystyle \int_{1}^{6}\frac{1}{x}dx \ = \ \frac{5}{48}[1+64/21+32/26+64/31+32/36+64/41+32/46+64/51+32/56+64/61\)

\(\displaystyle +32/66+64/71+32/76+64/81+32/86+64/91+16/96] \ \dot= \ 1.7920230584\)

\(\displaystyle Ergo, \ 1.78 \ \le \ \int_{1}^{6}\frac{1}{x}dx \ \le \ 1.80\)

\(\displaystyle A \ fixed \ n \ for \ Simpson's \ Rule \ usually \ gives \ a \ more \ accurate \ approximation \ than \ the\)

\(\displaystyle Trapezoid \ Rule, \ however \ the \ main \ reason \ we \ used \ the \ Trapezoid \ rule \ instead \ of \ the\)

\(\displaystyle Simpson \ Rule \ is \ its \ error \ can \ more \ easily \ be \ estimated.\)

\(\displaystyle For \ instance, \ if \ f(x) \ = \ \sqrt{xsin(x+1)}, \ then \ to \ estimate \ the \ error \ in \ Simpson's \ Rule\)

\(\displaystyle we \ would \ need \ to \ find \ the \ fourth \ derivative \ of \ f \ - \ a \ huge \ task!\)