Can someone explain integrals to an idiot...

Can someone explain integrals to an idiot? I know it is the area under the certain area of a graph but how is it calculated and how is it the opposite of a derivative?

Other urls found in this thread:

reddit.com/r/explainlikeimfive/comments/1d4eei/eli5_what_are_integrals_and_derivatives_in/
simple.wikipedia.org/wiki/Integral
twitter.com/NSFWRedditImage

The reason it's the opposite of a derivative is because the height of the graph (aka [math] f(x) [/math]) is the rate of area accumulation. So the integral [eqn] \int\limits_{0}^{x}f(t)\text{d}t [/eqn]
has a rate of change equal to the value of [math] f(x) [/math].

That's just bullshit semantics.

The real answer is the main theorem of integral and differential calculus, riemann and newton integral equivalence theorem and the definition of each of those. There is more to be told here, but I'm alredy bored explaining it for the umpteenth time. There's a lot about this everywhere on the internet.

Just read chapter 6 of the first Rudin

ill kill myself immidately after making this post but reddit (explain it like im 5) or simple english is good for stuff like this.

reddit.com/r/explainlikeimfive/comments/1d4eei/eli5_what_are_integrals_and_derivatives_in/

simple.wikipedia.org/wiki/Integral

This.
Integrals were never intuitive to me unlike derivatives.

An integral actually is simplification of the Riemann Sum expression.
>pic related
What you want to know is how the Riemann sum works. It uses a limit to add an infinite number of really small products (multiplications). The weird big E looking thing is the greek letter sigma -> s for sum. The way we know how to calculate such things with infinity by hand is because it is the opposite of the derivative, which we find from the Euler's method for approximating a point on a graph using the derivative at a known point. Euler's method of approximation: f(unkown b) = f(known a) + Riemann Sum. By subtracting f(known a) from both sides and substituting the integral from a to b of the derivative of f(x), we get the f(b) - f(a) = the integral from a to b of the derivative of f(x). Now if we just call the derivative of f(x) (or f'(x)) just plain f(x) then we know that f(x) from what was f(b) - f(a) must be functions whose derivative is f(x), or antiderivatives of f(x). Hopes this helps. I can go into more detail if this is bad. Also i have no idea how to use the equation tags sorry.

You realize OP asked for a "for dummies" (aka appeal to intuition) explanation, you fucking sperg.

There exists at least 20 different integrals that approach the same notion differently. The most common ones are the Lebesgue integral (the most relevant one for physicists) and the Riemann integral (less general than the Lebesgue integral, but gives the same values and is much easier to construct).

The idea behind the Lebesgue integral is that you have a function called a measure which gives you the "length" of a subset of [math]\mathbf R[/math], and you approximate your function better and better by functions that are pointwise smaller than it and take a finite number of different values (for these simple functions, defining an integral with the measure is trivial). The Lebesgue integral is the smallest upper bound of all the values you can get with these approximations.

The idea behind the Riemann integral is the rectangle approximation. You subdivide your interval into smaller and smaller elements, and for every element, you know in advance the integral of your function is bounded between the biggest rectangle in that element staying below the function curve, and the smallest rectangle staying above it. If the function is continuous almost everywhere (i.e. everywhere except on a set whose measure is 0, for example on a finite number of points), both bounds will converge towards the same value, and that value is the Riemann integral.

For continuous functions, integrals are calculated with the counter-derivative no matter what integral (not just Riemann or Lebesgue) you use. The fundamental theorem of calculus only requires the integral you use to verify these:
[eqn]\int_a^b \left[ f\left( x \right) \,+\, \lambda\, g\left( x \right) \right] \,\mathrm dx \,=\, \int_a^b f\left( x \right)\, \mathrm dx \,+\, \lambda\, \int_a^b g\left( x \right) \mathrm dx[/eqn]
[eqn]\left| \int_a^b f\left( x \right) \,\mathrm dx \right| \,\leqslant\, \int_a^b \left| f\left( x \right) \right|\, \mathrm dx[/eqn]
which are properties all integrals are expected to meet.

That's a good one lmao