Difference between first and second fundamental theorem of calculusIn first fundamental theorem of calculus,it states if A ( x ) = ∫ a x f ( t ) d t then A ′ ( x ) = f ( x ).But in second they say ∫ a b f ( t ) d t = F ( b ) − F ( a ),But if we put x=b in the first one we get A(b).Then what is the difference between these two and how do we prove A(b)=F(b)−F(a)?

Question

Difference between first and second fundamental theorem of calculusIn first fundamental theorem of calculus,it states if   A  (  x  )  =      ∫          a              x        f  (  t  )  d  t then       A    ′    (  x  )  =  f  (  x  ).But in second they say       ∫          a              b        f  (  t  )  d  t  =  F  (  b  )  −  F  (  a  ),But if we put x=b in the first one we get A(b).Then what is the difference between these two and how do we prove A(b)=F(b)−F(a)?

Bestvinajw · Accepted Answer

They have different assumptions.In the first part you mentioned, f is assumed to be continuous. In the second part, f can be assumed only Riemann integrable on the closed interval [a,b]. When f is continuous, the second part indeed follows from the first part.

Answered question

Answer & Explanation

New Questions in Integral Calculus