I can’t conceptually grasp this the same why I can grasp addition/subtraction (the opposite of adding stuff is to take it away), but if the derivative of a function is the slope at any given point, then why would the area under the curve be the opposite of that?
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It is quite hard to describe in text without diagrams, but there is a good video series on this very question by 3blue1brown on youtube: https://www.youtube.com/watch?v=rfG8ce4nNh0
in essence, the question of an integral is to answer “which derivative do i need to get such and such expression”, and it just turns out that this answer is related to the area under the curve.
From my other answer to same question:
Slope indicates how rapidly the function changes its value as it goes on.
Derivative of the integral (i.e area) describes how rapidly the area grows, i.e. the initial function itself (since the larger the function value is, the more it adds to the area).
Integral of the derivative means adding all those little slopes together. At every point the slope points to where the function is going next, so integrating them will, again, trace the initial function.
You get the slope by dividing the change in the function by a very tiny “width” of input.
You get an integral by multiplying the continuously changing value of the function by a very tiny “width” of input.
Both operations are either multiplying or dividing the function by an infinitesimally small “width” of input and hence are inverses to each other, like how you multiply by 5 to reverse dividing by 5.
I think people forget two things: (1) that the integral is a continuous analog of a sum (it’s a long S), and (2) it’s not taking the sum of a function f(x), but the sum of f(x) times some really small value dx. Think of f(x) as a height and dx as a width of a very skinny rectangle. The integral then is the sum of the areas of these skinny rectangles, which is the total area under the curve. And at any value x, that’s the amount of area added to the total area.
Now, the derivative of the integral is the rate of change this total area over a very tiny change in x. That rate of change is the area of the skinny rectangle at x. That area is f(x)dx. Divide that over the very tiny change of x, and you get f(x)dx/dx=f(x). That’s the original function.
You can also think of f(x)dx as the total differential (total incremental amount) in the area under the curve that is added on at x. So at each x, the rate at which area is added is f(x). Generally I find differentials more intuitive than derivatives when thinking about this. The derivative isn’t really the slope, but a rate of change in something. The derivative of the integral is the rate of change in the integral, which is the rate of change in the area, which is the height of the f(x)dx rectangle at x.
Let’s talk in terms of position and velocity since that is something most folks find intuitive.
When we take the derivative of the position function, we’re basically finding out the rate of change of position over time… but we’re doing that work as a limit process to find the exact rate-of-change at any particular infinitesimal slice of time. In the end we go from units of “distance” in our position function, to units of “distance-over-time” in our velocity graph.
So, how would we go in reverse- from a velocity graph to a position graph? Well, the units go from “distance-over-time” to just “distance” so we must be multiplying by “time” to make that happen. This is the same math as *”I drove 60mi/hr for 2hr… so I must have driven 120mi!”*… but we’re doing that work as a limit process to find the exact distance traversed over any particular infinitesimal slice of time.
Division is the opposite of multiplication.
Differentiation involves taking a thing, finding a tiny change in how far up it goes (B), a tiny change in how far along it goes (A), and dividing them to get the derivative (C = B/A).
Integration involves taking a thing (C), multiplying it by a tiny change in along (A), and getting the “area” or integral (B = A * C).
Ignoring the fancy infinitesimals and finite sums stuff, we have the same equation, just one using division, one using multiplication.
We can go more into this; if we increase the upper limit on an integral, we’re adding on an extra slice – that slice will have a “height” of the value of the function at that point. So the rate of change of our integral is just the function.
For differentiation, if we try to find the area under our rate of change curve, it will give us an infinite sum of the infinitely small changes in our function as we move along. Well if we add together how much our function has gone up by from some starting point to some end point, it will give us the actual value at that end point (up to a constant).
It’s the same as you said about addition and subtraction.
The derivative is a slope, calculated by taking the value of a function, **subtracting** a nearby value and **dividing** by the short horizontal distance between them. The integral is a cumulative area, calculated by **multiplying** a short horizontal distance by the value of the function and **adding** it to the total.
It’s the opposite arithmetic steps, in the opposite order. Everything else, limits and epsilons and all that, is just mathematicians being fancy. 😉
Let’s first consider the discrete analogue of some operators on sequences.
For a given sequence
>s = s_1, s_2, s_3, . . .
we can form a sequence *P(s)* of partial sums of the first one, two, three etc. terms, like this
>P(s) = s_1, s_1+s_2, s_1+s_2+s_3, . . .
and a sequence *D(s)* of differences between each term and the preceding term, like this
>D(s) = s_1, s_2-s_1, s_3-s_2, . . .
Now let’s compute *D(P(s)),* the result of applying both to some sequence *s.* We have
>P(s) = s_1, s_1+s_2, s_1+s_2+s_3, . . .
and for this sequence we list the sequence of differences between terms by definition of *D*
>D(P(s)) = s_1, (s_1+s_2)-(s_1), (s_1+s_2+s_3)-(s_1+s_2), . . .
but that is the original sequence *s = s_1, s_2, s_3, . . .* , so we observe that *D(P(s))=s.*
Similarly, for *P(D(s))* we have
>D(s) = s_1, s_2-s_1, s_3-s_2, . . .
so that
>P(D(s)) = s_1, s_1+(s_2-s_1), s_1+(s_2-s_1)+(s_3-s_2)
is again the original sequence. Thus also *P(D(s))=s*.
The operations *P* and *D* are therefore inverses of each other.
Finally, integrals and derivatives are similar operators but on *functions* instead of sequences. The difference is that while sequences have a clear “unit step”, there is no such thing for (real) functions, that’s what the *dx* in the integral and 𝛥*x* in the definition of the derivative represent and the rest is just some rigor, but the basic idea is the same.
Integral is kind of a “continuous sum” like the operator *P* and derivative is kind of a “continuous consecutive difference” like the operator *D.*
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