Are errors important in calculating integrals in Python?

[Samuel Rodriguez]

I am reading Mark Newman's Computational Physics textbook. He goes over calculating integrals with Simpsons's Rule and the Trapezoidal Rule, and then he goes over calculating their errors. Why would I have to ever worry about the error of the integral?

He has the chapters online at his website:

http://www-personal.umich.edu/~mejn/cp/chapters.html

He begins to go over this in chapter 5 on page 150.

 

[Incand]

Why wouldn't you worry about it? For example you can prove that the midpoint method is actually better (in the sense it's accurate to a higher order) than the trapezoid rule even through the order of the method itself is higher for the trapezoid method which would be somewhat counter-intuitive.

Furthermore when actually implementing a method you really need to estimate the error or you don't know if the result you get is accurate at all. Functions that change rapidly are much harder to integrate so you need a smaller step size (or perhaps a different method). Often you want the method to be accurate to a certain number of decimals for a specific problem.

 

[Samuel Rodriguez]

Oh I see. I was thinking of simple integrals ∫x²+ 2 dx lol. I wasn't thinking of harder integrals.

 

[Mark44]

Oh I see. I was thinking of simple integrals ∫x²+ 2 dx lol. I wasn't thinking of harder integrals.

The material you're reading has to do with definite integrals, such as $\int_1^3 x^2 + 2 dx$, not indefinite integrals.

There's a huge difference between finding a symbolic antiderivative (as in $\int x^2 + 2 dx$), and using numerical techniques to estimate a definite integral. The techniques mentioned in this thread are used to approximate definite integrals.

 

[Samuel Rodriguez]

The material you're reading has to do with definite integrals, such as $\int_1^3 x^2 + 2 dx$, not indefinite integrals.

There's a huge difference between finding a symbolic antiderivative (as in $\int x^2 + 2 dx$), and using numerical techniques to estimate a definite integral. The techniques mentioned in this thread are used to approximate definite integrals.

I understand you. It makes more sense. Thank you.