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Fundamental Theorem of Integral Calculus

Theorem: Let f be continuous function defined on the closed interval [a, b] and F be an anti- derivative of f, then 98581.png = [F(x)]ab = F(b) – F(a), where a and b are called the limits of integration, a being a lower or inferior and b being the upper or superior limit.

 

Notes:
  • If f(x) is not defined at x = a and x = band defined in open interval (ab) then 99079.png can be evaluated.
  • If 99073.png then the equation f(x) = 0 has at least one root lying in (ab) provided f is a continuous function in (a,b).
  • In 99064.png, the function f needs to be well defined and continuous in [ab].
     
    For instance, the consideration of definite integral 99058.png dx is erroneous since the function f expressed by f(x) = 99052.png is not defined in a portion –1 <x < 1 of the closed interval [–2, 3].




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