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My hope is to personally develop some further intuition for taking an integral (measuring the area under a curve). Consider a normal distribution and I need the area under the curve from $a$ to $b$. I know from calculus that the answer is given by:

$$P(a\le X \le b) = \int_{a}^{b} f(x)dx = \int_{a}^{b}\frac{1}{\sigma\sqrt{2\pi}}e^{−(y−\mu)^2/ 2\sigma^2} dx$$

My class instructor then draws a normal curve, indicates $a$ and $b$ on the horizontal axis (number line) and draws a line up from each point $a, b$ to the density function, connects the two crossing points and showing a square asks us, "How do we get the area of a square?" (Answer: base times height.)

It is then shown that the area of the square underestimates the area under the curve and then to get a better approximation the squares are redrawn as two rectangles and then four rectangles and then eight rectangles and this process shows that the area of the (smaller and smaller width) rectangles approximates the area under the curve better and better.

Next the instructor said that the "$f(x)$" part can be thought of as the height of the rectangle and the "$dx$" part can be thought of as the base (width) of the rectangle and that we want the base to be really small, in fact, infinitely small. The instructor then says something like, "Taking an integral or measuring the are under a curve is like summing the areas of rectangles with infinitely small width."

My questions:

  1. Are there other intuitive explanations of what is happening when we take an integral out there and would you please provide them?

  2. How would a pure mathematician explain an integral?

  3. Would the explanations (intuitive and mathematical) be fully consistent?

Multiple explanations or points of view would be appreciated.

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There are different ways to define integrals named after different people. What you teacher described is an informal explanation of the Riemann integral. You can see rigorous construction under the link, but it amounts to subdividing the interval of integration into subintervals of smaller and smaller lengths, and replacing the area under the graph with the sum of areas of the rectangles. The heights of the rectangles are equal to values of the function at some point on the subinterval, the sums are called Riemann sums. If as the sizes of the subintervals get uniformly smaller there exists a limit of the Riemann sums then the function is called Riemann integrable

This works well for continuous functions and others, but not for unbounded functions because then you can make some rectangles have arbitrarily large areas. For such cases a more general notion of Lebesgue integral is used. It is much more involved, but roughly instead of subdividing the integration interval you are subdividing the range of the integrated function into small subintervals, and then add up areas of "rectangles" with bases being sets where the function takes values from them. If the limit exists it is called the Lebesgue integral, and the function is called Lebesgue integrable. Every Riemann integrable function is Lebesgue integrable but not vice versa. Moreover, the base sets involved can be very complicated, and a whole Lebesgue measure theory has to be developed first to figure out their "lenghts".

Another way to define integral, called Daniell integral, is to approximate general functions by some "elementary" functions, whose integrals are either easy to compute as with step functions, or are already defined by some other construction, Riemann's for example. The integral is defined as the limit of integrals of approximating "elementary" functions. Daniell integral is equivalent to the Lebesgue integral in the sense that the same functions are integrable, and the values of integrals are the same. But it does not require developing measure theory in advance. There is a weaker but simpler version of Daniell integral called regulated integral.

There are also other more involved constructions like Henstock–Kurzweil integral, which is even more general than Lebesgue integral, Darboux integral, etc., but they are variations on the three ideas described above.

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One intuition (perhaps idiosyncratic) I have about integration comes from motion. Suppose we know how fast someone is going over a range of times; in that case we can work out how far they went. Intuitively, you could say that you take a lot of estimates of velocity $times$ distance at different times and add them up. But mathematically what you're doing is integrating the speed to get the distance.

One could also go the other way: If the speed itself is changing, then we can use the acceleration (the rate of change of speed) to find how much the velocity changes in a given time interval.

The way I like to explain this to intro. physics students: Integral calculus is a machine for turning information at earlier times (where they were, how fast they were moving) into information at future times. It takes the acceleration and spits out the velocity and position.

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