For a while I've been struggling with why the antiderivative should give the area under a curve, and a little more generally, how exactly integration and differentiation are inverses.
So recently, in learning more about the applications of integrals (specifically to do with volumes), the notion that what we're doing with the integral to find volume is adding up infinitely many 'slices' of the solid was explained to me. And seeing as taking a derivative is looking at a curve's slope point-by-point (broken up), my question is:
Is the intuition of going from 2D to 3D and vice versa a good way to think about the relationship between integrals and derivatives? Especially since the summing-up bit about integrals is what makes the solid, well solid. It also seems to reflect the loss of information in taking the derivative, in the way that when you only look at one dimension of a two dimensional shape, you don't get the whole idea.
Any corrections or thoughts would be helpful. Thanks!
EDIT It seems like the dimension bit isn't so clear, so I'll try to give more info. See, when my class was learning about volumes, the way my teacher described what the integral was in a way similar to this: since it's an infinite sum, it's taking all the slivers of area and adding them up to get a full solid. Thinking about integrals in that light, I focused on how we use 2D area of a slice, and integrate it to get the 3D volume of a solid. This put the idea in my brain that the process of integrating is similar to adding another dimension to a function. (in this case, integrating gave us a z-axis, kind of)
In that same thought process, differentiation can easily be thought of as the reverse: Taking a 2D curve and the process of homing in on one dimension, just a line. Also, I'll put a little more into my note on losing info while taking a derivative. As I said above, differentiation is like losing a dimension, or an axis in a way. So when we try to bring back another dimension through antidifferentiation, it's downright impossible to know where our line was in reference to the other axis without more information, since we dropped the axis and all the information it contained. That's why indefinite integrals require the constant added on: to make up for the lost y-axis information. I hope I cleared that up more, and thanks for any respones. :)