A ( x ) = ∫ a f x dx ... ( ) Based on this definition, the two basic fundamental theorems have been given. However, we only state them as their proofs are beyond the scope of this text book. .
. First fundamental theorem of integral calculus Theorem Let f be a continuous function on the closed interval [ a , b ] and let A ( x ) be the area function. Then A ′′′′′ ( x ) = f ( x ), for all x ∈∈∈∈∈ [ a , b ] . Fig .
. . Second fundamental theorem of integral calculus We state below an important theorem which enables us to evaluate definite integrals by making use of anti derivative. Theorem Let f be continuous function defined on the closed interval [ a , b ] and F be