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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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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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.

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).

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.

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

They are closer to _inverses_ than _opposites._ They somewhat undo each other.

Consider an example like this:

You drive 50 km/hr (the rate of change / tangent part) for 2 hrs (the bounds of the integral, 0 to 2). You traveled 100km (the area under the curve).

“Fifty kilometers per hour” and “One hundred kilometers” are not opposites.

(disclaimer: _inverse functions_ are unrelated to this)

They are not at all mathematically opposite in any way. Well… let me back up.

The derivative is the instantaneous rate of change of a function.

The definite integral is the bounded area under a curve of a function.

The indefinite integral is the family of antiderivatives of a function.

It’s a shame we called both an “integral” because there is nothing fundamentally connecting those two concepts. One is an area, the other is a family of functions. It just so happens that someone discovered that we can calculate the definite integral using the indefinite integral.

If you want to dive really deep though, look into Stoke’s theorem, the multi-dimensional version of the fundamental theorem of calculus. It basically says that the cumulative area of a function in a region can be determined by the value of the antiderivative of the function on the boundary of the region. Crazy stuff.

It’s probably easiest to break it down in terms of simple geometry.

Let’s draw two points. Now, write the equation for a slope between those two points:
(Horizontal Distance Between Points) / (Vertical Distance Between Points)

Now, write the equation for a rectangle that has those two points at opposite corners:
(Horizontal Distance Between Points) * (Vertical Distance Between Points)

In other words, the first (the derivative) is X / Y while the second (the integral) is X * Y. Does this make it easier to understand why they’re inverse operations?

The slope tells you whether the line is going to cover more or less area per x moving forward. If the rate of area coverage increases moving forward, then the line’s slope must be positive.

Both are characteristics of the same function. They are related, but they are not the same thing.

For example, if you are on a trip, it is possible to plot the velocity versus the time (how fast you are going at each moment).

The area under the curve is how far you traveled. The slope of the curve is how fast you are accelerating at that moment.

Different information/characteristics of the same thing.

Imagine a graph of a cars speed over time. The speed at any point in the graph is the cars speed at that moment. Now imagine a graph of that same car’s distance travelled. The sum of all the speed values ends up being the distance. (If you’re going 100 kilometers an hour, you’ll have travelled 100 kilometers after an hour.) The current distance is always the sum of all the speed so far.

Now think about it the other way. The change in distance is the speed. In other words, the slope of the distance line. The slope is a measure of how much things are changing, which is the derivative. If the slope of the distance is horizontal (zero) there is no speed. The steeper the slope, the more distance you’re adding at each step, which is to say speed.

It goes the same way for acceleration. Going back to the original graph of speed, if it’s completely horizontal there is no acceleration. But if the speed is increasing, the change (slope) is the acceleration. So if you graph acceleration it will be the derivative of speed. And speed is the integral of acceleration.

To get a derivative you need to subtract and divide (find difference and rate). To get an integral you need to multiply and add (find areas and add them together).

Responding to your edit. Indefinite integrals are basically defined as anti derivatives, rather than area under a curve. They are defined basically as the function when you take its derivative you end up back where you started. But this isn’t helpful conceptually as it’s just by definition.

But you can also express indefinite integrals as an area under the curve, but you set one of the bounds of the integral to be a variable x’ instead of a constant (the other bound you can just choose an arbitrary constant). Now the slope of this integral is really asking, how fast does the integrated function change with x, or in other words, how fast does the area under the original curve change with x? The additional area you add under the original curve f(x) is just going to be f(x) dx which brings you back to the original function.

Instead of starting at a function f(x) and asking, how is df/dx the opposite of integral f(x) dx, rather think about taking the slope of [ integral f(x) dx] and you see you end up where you started.

Think of it like we’re drawing a mountain. You place your pencil on the paper, and you move it left to right.

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The higher up you go, the more “massive” the mountain is that you’re drawing. Moving the straight across increases the mass of the mountain. However, if instead you move the pencil upwards and you move to the right, the mountain is getting even bigger, right?

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And if you move the pencil diagonally down, that would make the mountain less massive than if you had moved straight across.

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This is the intuition that shows the relationship between the slope of a curve, and the area underneath it. the “mass” of the “mountain” is the area.

Nice try. How many flugelbinders are in a doohickey

It used to drive me crazy in high school that the derivative of the volume of a sphere (four thirds pi r cubed) is the surface area of a sphere (four pi r squared). No teacher ever addressed this “coincidence.”

But after thinking about it for awhile I realized that if a sphere was growing, the tiniest little instantaneous additional volume being added would be the size of its surface.

That helped me a lot. Maybe it helps you too.

Homie I don’t think that question goes here. I don’t even understand what the question is

Someone needs to translate this question like I’m 5.

Opposite isn’t quite their right work. They are inverse functions. In an ideal work, if we have a function f(x), integral of the derivative of f(x) = derivative of the integral of f(x) = f(x). This is what people mean when they say that the derivative and integral are opposite. It’s the same idea that addition and subtraction, multiplication and division, or exponents and logarithms are opposites. Apply one to the other should get you the same result.

There is a problem though: the derivative is a “lossy” function. If we imagine a triangle and the same triangle on top of the box, the slope of the line at the top of the triangle is the same, so they would have the same derivative, but the derivative loses the y position of a curve, if we take the integral of that derivative, we would get the wrong result for one of the two curves.

I like the intuition that area under a curve is like summing the areas of a bunch of rectangle under the curve, ie. area = height * width. Then notice that slope is “rise over run” ie. slope = height / width. Then of course multiplying and dividing by width are “opposites” in the sense that they undo each other.

This is why the notation for integrals is “∫ f(x) dx)” where we multiply by “dx” (a small width of x) and the notation for derivatives is “df(x)/dx” where we divide by dx.

In this way we can see that area is the “opposite” of slope in the same way multiplication is the opposite of division.

If you take two values on the x-axis and vary the larger one, the rate of change of the area under the curve between them is the value of the function itself at that point.