A proof of the fundamental theorem of calculus

In summary, there are multiple rigorous versions of the proof of the fundamental theorem of calculus. The first to publish it was James Gregory in 1667, with the modern form being from Augustin L. Cauchy in 1823. It is also a special case of Stoke's theorem (1850) and Gauß' divergence theorem (1762). To further understand the rigorous versions of this proof, resources such as the book "The Real Numbers and Real Analysis" by Ethan Block can be helpful.
  • #1
Rishabh Narula
61
5
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.
 
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  • #2
Rishabh Narula said:
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).
 
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  • #3
fresh_42 said:
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?
 
  • #4
Rishabh Narula said:
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.
 
  • #5
fresh_42 said:
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.
 
  • #7
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.
 

FAQ: A proof of the fundamental theorem of calculus

What is the fundamental theorem of calculus?

The fundamental theorem of calculus is a mathematical theorem that establishes the relationship between differentiation and integration. It states that if a function is continuous on an interval and has an antiderivative, then the definite integral of the function over that interval is equal to the difference between the values of the antiderivative at the endpoints of the interval.

Why is the fundamental theorem of calculus important?

The fundamental theorem of calculus is important because it provides a powerful tool for evaluating definite integrals and solving real-world problems in fields such as physics, engineering, and economics. It also serves as the basis for many other important theorems in calculus.

What is the difference between the first and second parts of the fundamental theorem of calculus?

The first part of the fundamental theorem of calculus, also known as the fundamental theorem of calculus (FTC) 1, states that if a function f is continuous on an interval [a, b] and F is any antiderivative of f, then the definite integral of f from a to b is equal to F(b) - F(a). The second part, also known as FTC 2, states that if a function F is continuous on an interval [a, b] and f is its derivative, then the definite integral of f from a to b is equal to F(b) - F(a).

What is the relationship between the fundamental theorem of calculus and the chain rule?

The fundamental theorem of calculus and the chain rule are closely related. The chain rule is a rule for finding the derivative of a composite function, while the fundamental theorem of calculus is a rule for evaluating definite integrals. The second part of the fundamental theorem of calculus can be thought of as the inverse of the chain rule, as it allows us to find the original function when given its derivative.

Are there any limitations to the fundamental theorem of calculus?

Yes, there are some limitations to the fundamental theorem of calculus. It only applies to continuous functions that have antiderivatives. It also does not provide a method for finding the antiderivative of a function, which can sometimes be a difficult task. Additionally, the fundamental theorem of calculus assumes that the endpoints of the interval are fixed, which may not always be the case in real-world applications.

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