121616949-math.167

# 121616949-math.167 - F ± x = f x and G ± x = g x then Z b...

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7.3 Some Properties of Integrals 153 assumed that we were working left to right, but could as well number the subintervals from right to left, so that t 0 = b and t n = a . Then Δ t = t i +1 - t i is negative and in Z 5 6 v ( t ) dt = n - 1 X i =0 v ( t i t, the values v ( t i ) are negative but also Δ t is negative, so all terms are positive again. On the other hand, in Z 0 5 v ( t ) dt = n - 1 X i =0 v ( t i t, the values v ( t i ) are positive but Δ t is negative,and we get a negative result: Z 0 5 v ( t ) dt = - t 3 3 + 5 2 t 2 fl fl fl fl 0 5 = 0 - - 5 3 3 - 5 2 5 2 = - 125 6 . Finally we note one simple property of integrals: Z b a f ( x ) + g ( x ) dx = Z b a f ( x ) dx + Z b a g ( x ) dx. This is easy to understand once you recall that ( F ( x ) + G ( x )) = F ( x ) + G ( x ). Hence, if
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Unformatted text preview: F ± ( x ) = f ( x ) and G ± ( x ) = g ( x ), then Z b a f ( x ) + g ( x ) dx = ( F ( x ) + G ( x )) | b a = F ( b ) + G ( b )-F ( a )-G ( a ) = F ( b )-F ( a ) + G ( b )-G ( a ) = F ( x ) | b a + G ( x ) | b a = Z b a f ( x ) dx + Z b a g ( x ) dx. In summary, we will frequently use these properties of integrals: Z b a f ( x ) dx = Z c a f ( x ) dx + Z b c f ( x ) dx Z b a f ( x ) + g ( x ) dx = Z b a f ( x ) dx + Z b a g ( x ) dx Z b a f ( x ) dx =-Z a b f ( x ) dx...
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