Chap7_Sec7 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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7 TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION
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There are two situations in which it is impossible to find the exact value of a definite integral. TECHNIQUES OF INTEGRATION
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TECHNIQUES OF INTEGRATION The first situation arises from the fact that, in order to evaluate using the Fundamental Theorem of Calculus (FTC), we need to know an antiderivative of f . ( ) b a f x dx
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TECHNIQUES OF INTEGRATION However, sometimes, it is difficult, or even impossible, to find an antiderivative (Section 7.5). For example, it is impossible to evaluate the following integrals exactly: 2 1 1 3 0 1 1 x e dx x dx - +
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TECHNIQUES OF INTEGRATION The second situation arises when the function is determined from a scientific experiment through instrument readings or collected data. There may be no formula for the function (as we will see in Example 5).
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TECHNIQUES OF INTEGRATION In both cases, we need to find approximate values of definite integrals.
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7.7 Approximate Integration In this section, we will learn: How to find approximate values of definite integrals. TECHNIQUES OF INTEGRATION
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APPROXIMATE INTEGRATION We already know one method for approximate integration. Recall that the definite integral is defined as a limit of Riemann sums. So, any Riemann sum could be used as an approximation to the integral.
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APPROXIMATE INTEGRATION If we divide [ a , b ] into n subintervals of equal length ∆ x = ( b a )/n, we have: where x i * is any point in the i th subinterval [ x i -1 , x i ]. 1 ( ) ( *) n b i a i f x dx f x x =
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L n APPROXIMATION If x i * is chosen to be the left endpoint of the interval, then x i * = x i -1 and we have: The approximation L n is called the left endpoint approximation. 1 1 ( ) ( ) n b n i a i f x dx L f x x - = = Equation 1
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If f ( x ) ≥ 0, the integral represents an area and Equation 1 represents an approximation of this area by the rectangles shown here. L n APPROXIMATION
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If we choose x i * to be the right endpoint, x i * = x i and we have: The approximation R n is called right endpoint approximation. Equation 2 1 ( ) ( ) n b n i a i f x dx R f x x = = R n APPROXIMATION
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APPROXIMATE INTEGRATION In Section 5.2, we also considered the case where x i * is chosen to be the midpoint of the subinterval [ x i -1 , x i ]. i x
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M n APPROXIMATION The figure shows the midpoint approximation M n .
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M n APPROXIMATION M n appears to be better than either L n or R n .
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THE MIDPOINT RULE where and 1 2 ( ) [ ( ) ( ) ... ( )] b n a n f x dx M x f x f x f x = ∆ + + + b a x n - ∆ = 1 1 1 2 ( ) midpoint of [ , ] i i i i i x x x x x - - = + =
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TRAPEZOIDAL RULE Another approximation—called the Trapezoidal Rule—results from averaging the approximations in Equations 1 and 2, as follows.
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TRAPEZOIDAL RULE [ ] [ ] 1 1 1 1 1 0 1 1 2 1 0 1 2 1 1 ( ) ( ) ( ) 2 ( ( ) ( )) 2 ( ( ) ( )) ( ( ) ( )) 2 ... ( ( ) ( )) ( ) 2 ( ) 2 ( ) 2 ... 2 ( ) ( ) n n b i i a i i n i i i n n n n f x dx f x x f x x
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This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

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Chap7_Sec7 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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