4
$\begingroup$

Derivatives seem easy to understand abstractly as the rate of change of something, higher order derivatives are the rate of change of the rate of change of something, and so on.

I, however, have trouble understanding what an integral is in a general sense. It can be thought of as the sum of the infinitely small rectangles in an shape, but what is it, with respect to the initial function? I don't really understand why integration is the inverse operation of derivation. either.

  • 1
    "It can be thought of as the sum of the infinitely small rectangles in an shape" - precisely. You haven't been introduced to the notion of "area under a curve"?2011-12-08
  • 0
    My analysis lecturer used to tell us that we should have been "running out of the lecture theatre screaming" when he proved that integration is the opposite of differentiation. It is, in many ways, a surprising result, and the proof isn't very basic (this was in a senior honours analysis course).2011-12-08
  • 1
    The fact that "integration is the inverse operation of derivation" is (i) not *quite* accurate, but let that pass; and (ii) actually *should* be somewhat surprising. It's an unexpected connection when you think about the problems that give rise to the concepts (the 'instantaneous velocity' problem for derivatives, and the 'area problem' for integrals).2011-12-08

4 Answers 4