I seriously don't get this. See problem below.
Use the error formulas to find n such that the error in the approximation of the definite integral is less than 0.000001 using a) trapezoidal rule and b) simpson's rule

Question

I seriously don't get this. See problem below.
Use the error formulas to find n such that the error in the approximation of the definite integral is less than 0.000001 using a) trapezoidal rule and b) simpson's rule

jennychan12 · Answer

$$\int\limits_{1}^{2} \frac{ 1 }{ x }dx$$
I know how to use the error formulas and stuff, but what are the directions asking for?

jennychan12 · Answer

@RadEn  @Hero  @ZeHanz  
anyone?

zehanz · Answer

Instead of directly calculating the integral, which involves looking for a primitive function of 1/x and applying the Fundamental Theorem of Calculus, and yields an exact answer, one can also approximate it. The Trapezoidal Rule and Simpsons's Rule are two different ways of approximating integrals. The are  normally used only if an exact calculation cannot be done, i.e. because it is not possible to find a primitive function of the integrand.

jennychan12 · Answer

ok.
can u help me with this problem?
i don't get the last part of their answer

zehanz · Answer

The n is the number of trapezoids used to approximate the integral. The more trapezoids you use, the better the approximation, so the smaller the error. The following formula for the error holds:$$E \le \frac{ (3-1)^3 }{ 12n^2 }|f''(1)|$$Here f''(1)=2, hence the 2 in the formula. Appearently it is requiered for the error to be smaller than 0.00001, so the question is: how many trapezoids do I need for that? As you can see in the error formula, the larger n gets, the smaller the error. A two times larger n yields a 4x smaller error, due to n² in the denominator. If we calculate everything in the formula, we get:$$\frac{ 16 }{ 12n^2 }<0.00001$$Or:$$\frac{ 4 }{ 3n^2 }<0.00001$$This is the same as:$$\frac{ 3n^2 }{ 4 }>100000 =>3n^2>400000 => n^2>\frac{ 400000 }{ 3 }$$So: $$n^2 > 133,333.33 => n > \sqrt{133,333.33}=365.15$$Because n is an integer and n > 365.15, taking n > 366 ensures the error small enough. The explanation for Simpsons' Rule goes along the same lines...

jennychan12 · Answer

oh ok. thanks. i just don't get words as easily as i get numbers.
so basically finding n?

zehanz · Answer

yes!

zehanz · Answer

Thanks for this question! I had almost forgotten about this stuff...