Integration

Does anyone still consider the region of the integrand between the definite limits, post C2?

I still but I think most people don't as it isn't really assessed past that point.

Do you mean finding areas and stuff? That's both C2 and C4

Integration with definite limits isn't just used in pure maths. It comes up a lot in S2, and I've used it in M2 a few times

To be honest, I have no clue what you're asking..

Sorry guys. I explained myself poorly. I do mean in the context of finding areas. Checking to see whether the curve crosses the axis in the interval.

Sorry guys. I explained myself poorly. I do mean in the context of finding areas. Checking to see whether the curve crosses the axis in the interval.

Sorry guys. I explained myself poorly. I do mean in the context of finding areas. Checking to see whether the curve crosses the axis in the interval.

Oh right. Well, I sure hope they do. If anyone ever writes $\displaystyle\int_{-1}^{1}\dfrac{1}{x^2}\ dx = -2$, I will weep.

Oh right. Well, I sure hope they do. If anyone ever writes $\displaystyle\int_{-1}^{1}\dfrac{1}{x^2}\ dx = -2$, I will weep.

Oh right. Well, I sure hope they do. If anyone ever writes $\displaystyle\int_{-1}^{1}\dfrac{1}{x^2}\ dx = -2$, I will weep.

I imagine writing that caused you to weep a bit

That's kind of what I mean. I feel people forget to test for convergence as well as consider crossings (if shaded area) post C2.

I didn't think people did improper integrals at A level, let a lone tests for convergence! All the same, a good understanding of calculus involves knowing when your tools are appropriate, and being a good mathematician in general is checking your work makes sense! I feel they should be throwing these sorts of things into the papers!

It is possible I'm misremembering the syllabus. It is a shade after midnight

Well how would you write that?

New question.

What happens if you integrate across a discontinuity, in general, as opposed to breaking it up?
For example

$\displaystyle \int_0^e \dfrac{1}{x}dx$. The obvious discontinuity occurs at 0. I tried consulting google but the responses were varied. I'll check again tomorrow but hopefully someone might help.

Also, if your limits don't fit into the anti derivative of the integrand because of the anti derivative's domain (like lnx for 1/x), can you use facts (like its symmetry) to evaluate the integral?

Well, rather advanced stuff, but integration doesn't really care too much about discontinuities themselves. The issue they bring is that sometimes they cause the integral to diverge (like my example above), but otherwise, one can simply "leave out" the points that cause trouble. Secondly, yeah I think so.. I'm certainly no pro at integration, and I don't have an example to hand.. (have you ever looked at The Proof is Trivial? There's TSR's integration talent! you'll probably find some examples in there).