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Unformatted text preview: EVALUATING DEFINITE INTEGRALS Although we could calculate some definite integrals, it was quite tedious and time-consuming. Sir Issac Newton, the creator of Calculus, found a much simpler way to evaluate definite integrals by using antiderivatives. It consists of two different theorems, namely the Fundamental Theorem of Calculus part 1 and 2. We start with the second part Theorem 1 (The Fundamental Theorem of Calculus 2 (Evaluation Theorem)) . If f is a continuous on the interval [ a,b ] , then Z b a f ( x ) dx = F ( b )- F ( a ) where F is any antiderivative of f , that is, F = f . Proof. We divide the interval [ a,b ] into n subintervals with endpoints x = a,x 1 ,x 2 , , x n = b and with length x = ( b- a ) /n . Let F be any antiderivative of f . We can express the total difference in the F values as the sum of the differences over subintervals: F ( b )- F ( a ) = F ( x n )- F ( x ) = [ F ( x n )- F ( x n- 1 )] + [ F ( x n- 1 ) + F ( x n- 2 )] + + [ F ( x 1 )- F ( x )]...
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