121616949-math.161 - applications of integrals that we have...

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7.2 The Fundamental Theorem of Calculus 147 A complete proof is a bit too involved to include here, but we will indicate how it goes. First, if we can prove the second version of the Fundamental Theorem, theorem 7.4 , then we can prove the first version from that: Proof of Theorem 7.3 . We know from theorem 7.4 that G ( x ) = Z x a f ( t ) dt is an antiderivative of f ( x ), and therefore any antiderivative F ( x ) of f ( x ) is of the form F ( x ) = G ( x ) + k . Then F ( b ) - F ( a ) = G ( b ) + k - ( G ( a ) + k ) = G ( b ) - G ( a ) = Z b a f ( t ) dt - Z a a f ( t ) dt. It is not hard to see that R a a f ( t ) dt = 0, so this means that F ( b ) - F ( a ) = Z b a f ( t ) dt, which is exactly what theorem 7.3 says. So the real job is to prove theorem 7.4 . We will sketch the proof, using some facts that we do not prove. First, the following identity is true of integrals: Z b a f ( t ) dt = Z c a f ( t ) dt + Z b c f ( t ) dt. This can be proved directly from the definition of the integral, that is, using the limits of sums. It is quite easy to see that it must be true by thinking of either of the two
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Unformatted text preview: applications of integrals that we have seen. It turns out that the identity is true no matter what c is, but it is easiest to think about the meaning when a ≤ c ≤ b . First, if f ( t ) represents a speed, then we know that the three integrals represent the distance traveled between time a and time b ; the distance traveled between time a and time c ; and the distance traveled between time c and time b . Clearly the sum of the latter two is equal to the first of these. Second, if f ( t ) represents the height of a curve, the three integrals represent the area under the curve between a and b ; the area under the curve between a and c ; and the area under the curve between c and b . Again it is clear from the geometry that the first is equal to the sum of the second and third....
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