TITLE: Integration being the opposite of differentiation?
QUESTION [12 upvotes]: I know that if we start with an original function and take one derivative, we get another function. If we take the integral of that new function we get the original function back. So I see how they are opposite operations but I don't see how they are opposite in the sense that the integral is supposed to be the area under the curve and the derivative is supposed to be the instantaneous slope. It seems like they are separate; deriving one function is the opposite of integrating it but when we look at it graphically it doesn't make sense. Also, if I start with some function, differentiate it and then take the integral I get the original function so how is that the area under the curve? Thanks.

REPLY [5 votes]: The derivative is the instantaneous rate of change.  The best way to think about the derivative is:
\begin{equation}
f(x + \Delta x) \approx f(x) + f'(x) \Delta x.
\end{equation}
The (second) fundamental theorem of Calculus says, intuitively, that "the total change is the sum of all the little changes".
Suppose we want to know how much a function $f$ changes across an interval $[a,b]$.  Chop up $[a,b]$ into tiny subintervals $[x_i,x_{i+1}]$ of width $\Delta x_i$.  Then
\begin{align}
\underbrace{f(b) - f(a)}_{\text{total change}} &= \sum_i \underbrace{f(x_{i+1}) - f(x_i)}_{\text{little change}} \\
&\approx \sum_i f'(x_i) \Delta x_i \\
&\approx \int_a^b f'(x) \, dx.
\end{align}
It seems plausible that, by chopping up $[a,b]$ into extremely tiny subintervals, we can make this approximation as close as we like.  It follows that the two quantities must be equal:
\begin{equation*}
f(b) - f(a) = \int_a^b f'(x) \, dx.
\end{equation*}
The geometric interpretations of the derivative and the integral, are not always the best way to think about them.