Where does integral concept come from?












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$begingroup$


I have been searching for days to find why Newton offered an anti-derivative function to find the area under the curve. What was the reason for searching such a function? Was it just mathematical curiosity or was there a solid reason for that?






calculus integration definite-integrals







share|cite|improve this question











$endgroup$












  • $begingroup$
    Are you asking for an explanation of the mathematical concept itself or a historical explanation of how Newton came upon it?
    $endgroup$
    – Eric Wofsey
    Dec 16 '18 at 6:06










  • $begingroup$
    I was watching a bunch of videos where it says that in order to find the area under the curve Newton started to think about an anti-derivative of that function. I am just trying to understand what was the reason for searching an anti-derivative?
    $endgroup$
    – Arif Rustamov
    Dec 16 '18 at 6:17






  • 1




    $begingroup$
    I would say the theory of integration in Newton's time was a development beyond the quadrature technique known to the ancient Greeks. His achievement -- with the concept of a derivative already understood -- was to observe that the area $A(x)$ under the "curve" $y = f(x)$ above the segment $[a,x]$ had the property $frac{dA}{dx}(x) = f(x)$. I don't know the precise "reason for searching" but it is clear that Newton was exceptional in terms of scientific observation.
    $endgroup$
    – RRL
    Dec 16 '18 at 6:21










  • $begingroup$
    Historically when searching for the area $A(f(x),[0,t])$ under $f(x)=x^n,x in [0,t]$ or $f(x)=frac{1}{1+x}$ it is natural to search how $A(f(x),[0,t])$ transforms under $t mapsto t+1$ or $t mapsto 2t$. When those transformations are fully understood they give an explicit formula for the area.
    $endgroup$
    – reuns
    Dec 16 '18 at 6:25








  • 1




    $begingroup$
    Newton was not the first mathematician to do so. Go back in the history, Cavalieri, Torricelli, and before ...
    $endgroup$
    – user376343
    Dec 16 '18 at 20:09
















1












$begingroup$


I have been searching for days to find why Newton offered an anti-derivative function to find the area under the curve. What was the reason for searching such a function? Was it just mathematical curiosity or was there a solid reason for that?






calculus integration definite-integrals







share|cite|improve this question











$endgroup$












  • $begingroup$
    Are you asking for an explanation of the mathematical concept itself or a historical explanation of how Newton came upon it?
    $endgroup$
    – Eric Wofsey
    Dec 16 '18 at 6:06










  • $begingroup$
    I was watching a bunch of videos where it says that in order to find the area under the curve Newton started to think about an anti-derivative of that function. I am just trying to understand what was the reason for searching an anti-derivative?
    $endgroup$
    – Arif Rustamov
    Dec 16 '18 at 6:17






  • 1




    $begingroup$
    I would say the theory of integration in Newton's time was a development beyond the quadrature technique known to the ancient Greeks. His achievement -- with the concept of a derivative already understood -- was to observe that the area $A(x)$ under the "curve" $y = f(x)$ above the segment $[a,x]$ had the property $frac{dA}{dx}(x) = f(x)$. I don't know the precise "reason for searching" but it is clear that Newton was exceptional in terms of scientific observation.
    $endgroup$
    – RRL
    Dec 16 '18 at 6:21










  • $begingroup$
    Historically when searching for the area $A(f(x),[0,t])$ under $f(x)=x^n,x in [0,t]$ or $f(x)=frac{1}{1+x}$ it is natural to search how $A(f(x),[0,t])$ transforms under $t mapsto t+1$ or $t mapsto 2t$. When those transformations are fully understood they give an explicit formula for the area.
    $endgroup$
    – reuns
    Dec 16 '18 at 6:25








  • 1




    $begingroup$
    Newton was not the first mathematician to do so. Go back in the history, Cavalieri, Torricelli, and before ...
    $endgroup$
    – user376343
    Dec 16 '18 at 20:09














1












1








1





$begingroup$


I have been searching for days to find why Newton offered an anti-derivative function to find the area under the curve. What was the reason for searching such a function? Was it just mathematical curiosity or was there a solid reason for that?






calculus integration definite-integrals







share|cite|improve this question











$endgroup$




I have been searching for days to find why Newton offered an anti-derivative function to find the area under the curve. What was the reason for searching such a function? Was it just mathematical curiosity or was there a solid reason for that?






calculus integration definite-integrals




calculus integration definite-integrals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 16 '18 at 6:07









Eric Wofsey

189k14216347




189k14216347










asked Dec 16 '18 at 6:04









Arif RustamovArif Rustamov

367




367












  • $begingroup$
    Are you asking for an explanation of the mathematical concept itself or a historical explanation of how Newton came upon it?
    $endgroup$
    – Eric Wofsey
    Dec 16 '18 at 6:06










  • $begingroup$
    I was watching a bunch of videos where it says that in order to find the area under the curve Newton started to think about an anti-derivative of that function. I am just trying to understand what was the reason for searching an anti-derivative?
    $endgroup$
    – Arif Rustamov
    Dec 16 '18 at 6:17






  • 1




    $begingroup$
    I would say the theory of integration in Newton's time was a development beyond the quadrature technique known to the ancient Greeks. His achievement -- with the concept of a derivative already understood -- was to observe that the area $A(x)$ under the "curve" $y = f(x)$ above the segment $[a,x]$ had the property $frac{dA}{dx}(x) = f(x)$. I don't know the precise "reason for searching" but it is clear that Newton was exceptional in terms of scientific observation.
    $endgroup$
    – RRL
    Dec 16 '18 at 6:21










  • $begingroup$
    Historically when searching for the area $A(f(x),[0,t])$ under $f(x)=x^n,x in [0,t]$ or $f(x)=frac{1}{1+x}$ it is natural to search how $A(f(x),[0,t])$ transforms under $t mapsto t+1$ or $t mapsto 2t$. When those transformations are fully understood they give an explicit formula for the area.
    $endgroup$
    – reuns
    Dec 16 '18 at 6:25








  • 1




    $begingroup$
    Newton was not the first mathematician to do so. Go back in the history, Cavalieri, Torricelli, and before ...
    $endgroup$
    – user376343
    Dec 16 '18 at 20:09


















  • $begingroup$
    Are you asking for an explanation of the mathematical concept itself or a historical explanation of how Newton came upon it?
    $endgroup$
    – Eric Wofsey
    Dec 16 '18 at 6:06










  • $begingroup$
    I was watching a bunch of videos where it says that in order to find the area under the curve Newton started to think about an anti-derivative of that function. I am just trying to understand what was the reason for searching an anti-derivative?
    $endgroup$
    – Arif Rustamov
    Dec 16 '18 at 6:17






  • 1




    $begingroup$
    I would say the theory of integration in Newton's time was a development beyond the quadrature technique known to the ancient Greeks. His achievement -- with the concept of a derivative already understood -- was to observe that the area $A(x)$ under the "curve" $y = f(x)$ above the segment $[a,x]$ had the property $frac{dA}{dx}(x) = f(x)$. I don't know the precise "reason for searching" but it is clear that Newton was exceptional in terms of scientific observation.
    $endgroup$
    – RRL
    Dec 16 '18 at 6:21










  • $begingroup$
    Historically when searching for the area $A(f(x),[0,t])$ under $f(x)=x^n,x in [0,t]$ or $f(x)=frac{1}{1+x}$ it is natural to search how $A(f(x),[0,t])$ transforms under $t mapsto t+1$ or $t mapsto 2t$. When those transformations are fully understood they give an explicit formula for the area.
    $endgroup$
    – reuns
    Dec 16 '18 at 6:25








  • 1




    $begingroup$
    Newton was not the first mathematician to do so. Go back in the history, Cavalieri, Torricelli, and before ...
    $endgroup$
    – user376343
    Dec 16 '18 at 20:09
















$begingroup$
Are you asking for an explanation of the mathematical concept itself or a historical explanation of how Newton came upon it?
$endgroup$
– Eric Wofsey
Dec 16 '18 at 6:06




$begingroup$
Are you asking for an explanation of the mathematical concept itself or a historical explanation of how Newton came upon it?
$endgroup$
– Eric Wofsey
Dec 16 '18 at 6:06












$begingroup$
I was watching a bunch of videos where it says that in order to find the area under the curve Newton started to think about an anti-derivative of that function. I am just trying to understand what was the reason for searching an anti-derivative?
$endgroup$
– Arif Rustamov
Dec 16 '18 at 6:17




$begingroup$
I was watching a bunch of videos where it says that in order to find the area under the curve Newton started to think about an anti-derivative of that function. I am just trying to understand what was the reason for searching an anti-derivative?
$endgroup$
– Arif Rustamov
Dec 16 '18 at 6:17




1




1




$begingroup$
I would say the theory of integration in Newton's time was a development beyond the quadrature technique known to the ancient Greeks. His achievement -- with the concept of a derivative already understood -- was to observe that the area $A(x)$ under the "curve" $y = f(x)$ above the segment $[a,x]$ had the property $frac{dA}{dx}(x) = f(x)$. I don't know the precise "reason for searching" but it is clear that Newton was exceptional in terms of scientific observation.
$endgroup$
– RRL
Dec 16 '18 at 6:21




$begingroup$
I would say the theory of integration in Newton's time was a development beyond the quadrature technique known to the ancient Greeks. His achievement -- with the concept of a derivative already understood -- was to observe that the area $A(x)$ under the "curve" $y = f(x)$ above the segment $[a,x]$ had the property $frac{dA}{dx}(x) = f(x)$. I don't know the precise "reason for searching" but it is clear that Newton was exceptional in terms of scientific observation.
$endgroup$
– RRL
Dec 16 '18 at 6:21












$begingroup$
Historically when searching for the area $A(f(x),[0,t])$ under $f(x)=x^n,x in [0,t]$ or $f(x)=frac{1}{1+x}$ it is natural to search how $A(f(x),[0,t])$ transforms under $t mapsto t+1$ or $t mapsto 2t$. When those transformations are fully understood they give an explicit formula for the area.
$endgroup$
– reuns
Dec 16 '18 at 6:25






$begingroup$
Historically when searching for the area $A(f(x),[0,t])$ under $f(x)=x^n,x in [0,t]$ or $f(x)=frac{1}{1+x}$ it is natural to search how $A(f(x),[0,t])$ transforms under $t mapsto t+1$ or $t mapsto 2t$. When those transformations are fully understood they give an explicit formula for the area.
$endgroup$
– reuns
Dec 16 '18 at 6:25






1




1




$begingroup$
Newton was not the first mathematician to do so. Go back in the history, Cavalieri, Torricelli, and before ...
$endgroup$
– user376343
Dec 16 '18 at 20:09




$begingroup$
Newton was not the first mathematician to do so. Go back in the history, Cavalieri, Torricelli, and before ...
$endgroup$
– user376343
Dec 16 '18 at 20:09










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