Fundamental Theorems Of Integral Calculus
Fundamental Theorems Of Integral Calculus : First Fundamental Theorem Of Integral Calculus : Theorem 1 : If a function f is bounded and integrable on [ a,b ] , then the function F defined as x F(x) = ∫ f(t) dt , a≤x≤b 0 is continuous on [ a,b ] and further more , if f is continuous at a point of [ a,b ] , then F is derivable at c and F'(c) = f(c) . Proof : It is given that the function f is bounded . Then by definition ∃ a number k such that |f(x)|≤ k for x ∈ [ a,b ] ......(1) Let x₁ , x₂ ∈[ a,b ] such that a≤x₁≤x₂≤b . x₂ x₁ Then |F(x₂) - F(x₁) | = |∫ f(t) dt - ∫ f(t) dt | a a x₂ a = |∫ f(t) dt + ∫ f(t) dt |