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My question is this, (it comes from the fact that i've kind of been skipping around in the books).

The definite integral of a curve from x1 to x2 gives you the Exact area under the curve right?

I only ask because the books both cover a lot of riemann sums and stuff, like breaking the curve up into smaller portions (rectangles) and such. Why would one do that when you can calculate the Exact area from the definite integral?

Or am i wrong, haha, sorry for the extremely basic question, i've only learned the basics of limits, derivatives and integrals.

Thanks for the help =)