if mathematically derivatives are the opposite of integrals, conceptually how is the area under a curve opposite to the slope of a tangent line?
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A simple example of this is with two basic physics functions—velocity and distance. Velocity being the derivative of distance, indicates the slope of that function, or the rate the distance changes with time. The integral of all of those rate changes ends up being the sum of all of the instantaneous velocity values over the course of a time period, to indicate the distance, or area under the velocity curve.
The relationships are easier to conceptualize looking at Riemann sums too
Derivatives and integrals are mathematically opposite, in calculus. To help you visualize, teachers may draw a parallel to some geometry concepts, but the geometry visualizations are not going to be conceptually opposite.
Unless you look at the geometry from a calculus point of view. Calculus studies “change” rather than fixed objects.
So if you imagine [a parabola](https://www.math.net/img/a/algebra/functions/quadratic-function/parabola.png), the derivative of it is not the slope of A tangent line, it’s what ALL the tangent lines do. It’s a function that describes the behavior of ALL tangent lines, which is that their slopes decrease to 0 and continue to decrease into the negative.
And the integral, “area under the curve”, again it’s not just the TOTAL area, it’s as you go along, from x=-2 to x=3 for example, it’s a description of what the function does, when you look at each point along the line (the area increases over time).
And that’s perhaps where your “opposite” hides: the slopes decrease and the area increases.
But in general, calculus is about systems that change, and trying to understand change with “pictures” (geometrical shapes are “fixed” in time) is detrimental, you can’t take the analogy very far, it loses too much in the translation.
Let’s say you have the derivative of a function. So, you know how the function varies.
In order for you to know the original function, you just need to integrate the derivative that is, add up little pieces that make the derivative, since they together “make” the original function
So, for instance, if you have a derivative and break it down into discrete intervals of 3 5 2 -1, that means it grew 3, then 5, then 2, then shrinked one. By integrating, you’ll add up all the area under the curve, so the cumulative effect of each variation, and get an original function of 0 3 8 10 9 (assuming you’re starting at 0).
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.
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