A proof of the fundamental theorem of calculus

[Rishabh Narula]

is there a rigorous version of this proof of fundamental theorem of calculus?if yes,what is it?and who came up with it?
i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it.
i know it must be known...but still I've been curious about it so just wanted to ask.

so here goes my proof...

imagine two curves.one is deravitive and the other its antideravitive.we want the area under the deravitive curve.and the fundamental theorem says that the area under the deravitive curve is the difference between two y values of the antideravitive curve.I simply justify that by saying...the area under the deravitive curve as mentioned in many books is sum of infnite tiny rectangles each of area f(x)dx.now this area of each rectangle is also equal to small change dy in antideravitive curve since dy of antideravitive curve is also instantaneous slope times dx or again f(x)dx.thus on left side of fundamental theorem we are adding up infinite small rectangle areas to get the total area...and on right handside we are calculating the sum of infinite dy s by finding the difference between two y values of antideravitive curve.

 

[fresh_42]

is there a rigorous version of this proof of fundamental theorem of calculus?

Of course. I even suspect: many!

if yes,what is it?and who came up with it?

"The first to publish this was in 1667 James Gregory in Geometriae pars universalis ... His modern form is from Augustin L. Cauchy (1823)." (Wikipedia)

i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it.
i know it must be known...but still I've been curious about it so just wanted to ask.

so here goes my proof...

imagine two curves.one is deravitive and the other its antideravitive.we want the area under the deravitive curve.and the fundamental theorem says that the area under the deravitive curve is the difference between two y values of the antideravitive curve.I simply justify that by saying...the area under the deravitive curve as mentioned in many books is sum of infnite tiny rectangles each of area f(x)dx.now this area of each rectangle is also equal to small change dy in antideravitive curve since dy of antideravitive curve is also instantaneous slope times dx or again f(x)dx.thus on left side of fundamental theorem we are adding up infinite small rectangle areas to get the total area...and on right handside we are calculating the sum of infinite dy s by finding the difference between two y values of antideravitive curve.

It is also a special case of Stoke's theorem (1850; modern version Cartan 1945) and Gauß' divergence theorem (Lagrange 1762; Gauß 1813; G. Green 1825; M. Ostrogradski 1831 with the first formal proof).

 

[Rishabh Narula]

Of course. I even suspect: many!

"The first to publish this was in 1667 James Gregory in Geometriae pars universalis ... His modern form is from Augustin L. Cauchy (1823)." (Wikipedia)

It is also a special case of Stoke's theorem (1850; modern version Cartan 1945) and Gauß' divergence theorem (Lagrange 1762; Gauß 1813; G. Green 1825; M. Ostrogradski 1831 with the first formal proof).

hey thanks for the answer.could you provide links too?

 

[fresh_42]

hey thanks for the answer.could you provide links too?

I have found all of them on some language version of Wikipedia. There are also links, but usually to some books with the corresponding theorems.

 

[Rishabh Narula]

I have found all of them on some language version of Wikipedia. There are also links, but usually to some books with the corresponding theorems.

oh,okay,i'll just google.would love to understand rigrous versions of this.

 

[PeroK]

oh,okay,i'll just google.would love to understand rigrous versions of this.

Try this to start:

http://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx#Extras_IntPf_FToCII

 

[member 587159]

I highly recommend the book "the real numbers and real analysis" by Ethan Block for the topic of Riemannintegration. It has the best exposition on the subject I have seen so far and has facts I didn't see in other textbooks.