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Basic Properties of the Integral Notes

# Basic Properties of the Integral Notes - The Basic...

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The Basic Properties of the Integral When we compute the derivative of a “complicated” function, like x 2 + sin x , we usually use diFerentiation rules, like d d x [ f ( x ) + g ( x )] = d d x f ( x ) + d d x g ( x ), to reduce the computation to that of ±nding the derivatives of the simple parts, like x 2 and sin x , of the complicated function. Exactly the same technique is used in computing integrals. Here are a bunch of integration rules. Let a, b and A, B, C be real numbers. Let the functions f ( x ) and g ( x ) be integrable on an interval that contains a and b . Then (a) i b a [ f ( x ) + g ( x )] dx = i b a f ( x ) dx + i b a g ( x ) dx (b) i b a [ f ( x ) - g ( x )] dx = i b a f ( x ) dx - i b a g ( x ) dx (c) i b a [ Cf ( x )] dx = C i b a f ( x ) dx (d) i b a [ Af ( x ) + Bg ( x )] dx = A i b a f ( x ) dx + B i b a g ( x ) dx That is, integrals depend linearly on the integrand. (e) i b a dx = b - a Theorem 1 (Arithmetic of Integration) . Proof. The ±rst three formulae are all special cases of formula (d). ²or example, formula (a) is just formula (d) with A = B = 1. So to get the ±rst four formulae it’s good enough to just prove formula (d), which we’ll do by just comparing the de±nitions of the left and right hand sides. Let’s introduce the notation R n ( h, a, b ) = n s i =1 h p a + ( i - 1 2 ) b - a n P b - a n (1) for the midpoint Riemann sum approximation to I b a h ( x ) dx with n subintervals. Then, by de±nition, the left hand side, namely I b a [ Af ( x ) + ( x )] dx , is the limit as n → ∞ of R n ( Af + Bg, a, b ) while the right hand side, namely A I b a f ( x ) dx + B I b a g ( x ) dx , is the limit as n → ∞ of AR n ( f, a, b ) + BR n ( g, a, b ). But R n ( Af + Bg, a, b ) = n s i =1 b Af p a + ( i - 1 2 ) b - a n P + p a + ( i - 1 2 ) b - a n PB b - a n = n s i =1 b Af p a + ( i - 1 2 ) b - a n P b - a n + p a + ( i - 1 2 ) b - a n P b - a n B 1 January 13, 2014

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= n s i =1 Af p a + ( i - 1 2 ) b - a n P b - a n + n s i =1 Bg p a + ( i - 1 2 ) b - a n P b - a n
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