Is Integration the inverse of differentiation?

[rampalli_aravind]

hi everybody,

I want to know if integration is the inverse process of differentiation?
Please explain in a detailed way.

Thank you and with regards.

 

[TD]

Well yes and no... Inuitively, they are inverse operations but formally, no (but the experts will probably get back to you) because ("indefinite") integration is only determined up to an extra arbitrary constant.

$$\frac{{dx^2 }} {{dx}} = 2x{\text{ but }}\int {2xdx = x^2 + C}$$

 

[rampalli_aravind]

thank you TD

well...thank you for the immediate reply

well...if differentiation is the inverse of integration, how can we justify tht they r inverses with respect to the geometric representation of both

thank you again

 

[Brad Barker]

well...thank you for the immediate reply

well...if differentiation is the inverse of integration, how can we justify tht they r inverses with respect to the geometric representation of both

thank you again

well, that certainly IS strange! when i learned calc, i found it really troubling. but now that it's been a few years, i am less bothered. :tongue:

anyway, the geometrical relationship between the two is the fundamental theorem of calculus!

the rate of change (wrt to the independant variable) of the area of a function is the function itself!

 

[TD]

Could you please specify what you mean with their geometric representation?
Such as a slope for the derivative and an area for the integral? (Although those aren't the only possible interpretations of course...)

 

[rampalli_aravind]

well...i want some detailed reply. i want to know the other geometrical interpretations of integration and deifferentiation. as far as i know abt the geo. interpretations, it is only SLOPE and AREA...and how can we have them as inverses?

 

[matt grime]

they are not inverse operations, differentiation, not being one to one cannot possibly be invertible. poeple often label integration as anti-differentiation, though.

as for why the fundamental theorem of calculus is true (that if f is continuous and F(x) is the integral of f from a to x where a is some constant, then F is differentible and the derivative is f) what it is saying geometrically is that if we tak f, this continuous function, and look at the rate at which the area it defines changes then that is f itself. which verbally seems quite reasonable.

but anyway, it is a formal consequence of the definition and as with most scientific results you won't bet very far if you use to many "causal" ideas.

then again i suppose we could explain it by giving a good definition of the derivative rather than just "the slope", ie one that is often times more useful and is how we ought to define it.

let f be a function, and suppse that it is differentiable, then the derivative f' is a function with the following property:

f(x+d) = f(x)+d f'(x) + junk that behaves like d^2 or worse.

ie if we change x by a small amount d then we near as damnit change f(x) by d times the derivative at x.

what is the integral of a function g from a to b? it is approximately g(a)d +g(a+d)d +g(a+2d)d+...+g(b)d

where we spit the interval fom a to b into lots of little subintervals each of width d and estimate by this sum. now surely you can see that as d gets small in both examples how it is that the derivative of the integral of g, or the integral of the derivative of f might be linked?

 

[rampalli_aravind]

i'll get back to u for further clarifications...thank you for those replies

and can u post me the geometrical interpretations of both differentiation and integration.

 

[TD]

Well, the most 'basic' ones:
- The derivative of f(x) in a point a gives the rate of change there -> the slope if you wish
- The integral of f(x) on the interval [a,b] gives the area between f(x) and the x-axis on that interval

This is 'very' basic of course, and probably not mathematically precise. But you understand the meaning?

 

[rampalli_aravind]

anyother interpretation?

i know those two interpretations

i want to know if there is anything else - regarding geometrical interpretations

please let me know

and i also want to know if there are some very good sets of problems in any website or book...

i want to master integration

pls take the trouble to list the websites/books.

thank you

 

[matt grime]

i'll get back to u for further clarifications...thank you for those replies

and can u post me the geometrical interpretations of both differentiation and integration.

i already did. reread my post and think about it.

 

[Brad Barker]

i know those two interpretations

i want to know if there is anything else - regarding geometrical interpretations

please let me know

and i also want to know if there are some very good sets of problems in any website or book...

i want to master integration

pls take the trouble to list the websites/books.

thank you

umm... james stewart's calculus: early transcendentals has a lot of stuff in it...