Why do some integrals require correction for extreme precision?

[bobie]

I have read that even the simplest integrals (like y=x²) might need some correction if we want to reach an extreme precision. Is that really so?
Can you explain why or give me some useful links?
Thanks

 

[pasmith]

I have read that even the simplest integrals (like y=x²) might need some correction if we want to reach an extreme precision. Is that really so?
Can you explain why or give me some useful links?
Thanks

Define "extreme precision".

Integrals which can be done analytically (such as the integral of $x^2$) cannot be more precise.

If you can only do an integral numerically then you are constrained by the fact that computers can only ever do a finite number of calculations on a finite subset of the rational numbers, so there's always going to be some error; the question is whether the error can be made small enough so that you can ignore it in the context of whatever your actual problem is.

 

[pbuk]

I have read that even the simplest integrals (like y=x²) might need some correction if we want to reach an extreme precision. Is that really so?
Can you explain why or give me some useful links?
Thanks

No. $\int_a^b x^2 \,dx = 2(b - a)$ there is no "correction" involved.

Can you explain why you think this or give us a useful link?

 

[collinsmark]

No. $\int_a^b x^2 \,dx = 2(b - a)$ there is no "correction" involved.

Don't you mean,

$$\int_a^bx^2 dx = \frac{1}{3}\left( b^3 - a^3 \right)?$$

 

[Mark44]

I have read that even the simplest integrals (like y=x²) might need some correction if we want to reach an extreme precision. Is that really so?

If you use a numerical technique, the results are generally imprecise. As already mentioned in this thread, if you are lucky enough to know the antiderivative of the function you're integrating, the results will be exact.

Can you explain why or give me some useful links?
Thanks

There are lots of numerical methods for integration - trapezoid rule, Simpson's rule, many others. See http://en.wikipedia.org/wiki/Numerical_integration for more information.

 

[pbuk]

Don't you mean,
$$\int_a^bx^2 dx = \frac{1}{3}\left( b^3 - a^3 \right)?$$

Oops yes of course, thank you - now how did that happen? :blush: