Update

calculus.md

@@ -176,7 +176,7 @@ As already claimed in the section on derivative, integrating is **more difficult
 So due to these complications we now yet have to explain the two different types of integrals:
 
 - **indefinite integral**: This is the FUNCTION we get by performing integration, i.e. result of indefinite integral is a mathematical expression with variables in it. In fact this expression represents an infinite set of functions because it always has the additive constant *C* in it (like hinted above) -- we can kind of ignore this for now. The important gist is this: indefinite integral kind of gives us a general FORMULA that can further be used to compute definite integrals. For example an indefinite integral of function *f(x) = 1* will be *x + C*. In practice the result we are searching is often a definite integral (a single value), but to compute that we have to start by computing the indefinite integral. However it's also very hard to calculate indefinite integrals -- they are the precise solution and holy grail of integration but in practice we can't always get them and have to resort to approximations.
-- **definite integral**: This is a single [NUMBER](number.md) which (applying the geometric interpretation of integral) tells us the AREA below the function graph (with area below zero counting as negative) over some specific INTERVAL, i.e. between two given points A and B. This means that definite integral doesn't give us an expression but rather a quantity. For example a definite integral of function *f(x) = 1* over interval [0,1] will give us 1 (imagine the graph: the area is simply that of a square with side 1). Definite integrals are computed from the indefinite integral by plugging the upper interval number into the indefinite integral (in the place of the variable), then plugging the lower interval number, and then subtracting the latter from the former. With numeric methods (computer integration) we always only get definite integrals (and actually only their approximate values) -- the computer here skips computing the indefinite integral (as that's hard) and rather like a dumb machine LITERALLY goes by small steps and computes the area below the function graph. Small note to this: the computer still can draw a graph of a function's integral by plotting definite integral value for interval 0 to *x* for every plotted *x* because when we think about it, the indefinite integral kind of gives us a function of how an indefinite integral grows; so the computer can give us a picture of a graph but it generally cannot give us an analytically computed formula of indefinite integral.
+- **definite integral**: This is a single [NUMBER](number.md) which (applying the geometric interpretation of integral) tells us the AREA below the function graph (with area below zero counting as negative) over some specific INTERVAL, i.e. between two given points A and B. This means that definite integral doesn't give us an expression but rather a quantity. For example a definite integral of function *f(x) = 1* over interval [0,1] will give us 1 (imagine the graph: the area is simply that of a square with side 1). Definite integrals are computed from the indefinite integral by plugging the upper interval number into the indefinite integral (in the place of the variable), then plugging the lower interval number, and then subtracting the latter from the former. With numeric methods (computer integration) we always only get definite integrals (and actually only their approximate values) -- the computer here skips computing the indefinite integral (as that's hard) and rather like a dull machine LITERALLY goes by small steps and computes the area below the function graph. Small note to this: the computer still can draw a graph of a function's integral by plotting definite integral value for interval 0 to *x* for every plotted *x* because when we think about it, the indefinite integral kind of gives us a function of how an indefinite integral grows; so the computer can give us a picture of a graph but it generally cannot give us an analytically computed formula of indefinite integral.
 
 Fun fact: before digital computers engineers used very clever methods to find definite integrals of general functions. [Analog](analog.md) computers were particularly good at integrating, their continuous nature makes them a quite elegant solution to the problem, however perhaps even more genius method in its [simplicity](kiss.md) was the following: the engineer would draw the function he wanted to integrate on a sheet of paper (or maybe more preferably some kind of heavier material), then cut it out and simply weight its mass -- this would give him the fraction of the weight of the whole sheet of paper and so also the fraction of the area below the function graph.