Is there a way to derive the formula of how the integral of a function yields the area under the graph?

Answer 1

It's called the Fundamental Theorem of Calculus, where once differentiation is defined and known, integration of functions for the area can be computed. The wiki article gives a good review of a couple of ways to derive it. Differentiating a function gives you a rate of change of the value of the function, so let f(x) be the function that we wish to integrate. Let g(x) be the integral of f(x), which expresses the area under f(x) from, say, 0 to x. Then the rate change in g(x) at x depends on how f(x) changes at x, or g'(x) = f(x). The inverse of this equation is g(x) = S( f(x) ) where S is the integral sign. S is the inverse of the differential operator d/dx. Be careful to note the difference between "rate of change", and "change", those are not the same thing. For example, if I am driving, a change in distance could be "one mile", but a rate of change of distance could be "one mile a minute".

Answer 2

Yes, often there is a way, and it depends entirely on the equation of the function. So, functions like f(x) = x^2, or f(x) = sin(x) give rise to integrals with nice solutions; but many really interesting functions do not have simple integrals.

HTH

Charles

Answer 3

permit's say a = acceleration = m/s^2 Integrating that yields speed. v = speed = m/s Integrating speed yields distance from the beginning. s = distance from beginning = m. the 2nd fundamental is the section decrease than the curve of the 1st fundamental function.

Answer 4

yep....the rienmann sum.....

u use sigma of all the f(x) in each x multiplied by the x interval u place.....

or the definition of an integral...which is the same as the above except the divisions have actual numerical values....

This probably duznt make sense if u dont know this stuff....u will take it in calculus.....it needs a couple of sessions or so...

Answer 5

This is what differential calculus is all about

Answer 6

RIEMAN SUMS

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