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Chap7_Sec7

# 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
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 - +
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.
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.
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
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
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 .
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 - - = + =
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 x f x f x x f x f x f x f x f x f x x f x f x f x f x f x - = = - = - - ∆ + = + = + + + + + + = + + + + +
THE TRAPEZOIDAL RULE where ∆ x = ( b a )/ n and x i = a + i x [ ] 0 1 2 1 ( ) ( ) 2 ( ) 2 ( ) 2 ... 2 ( ) ( ) b n a n n f x dx T x f x f x f x f x f x - = + + + + +

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TRAPEZOIDAL RULE The reason for the name can be seen from the figure, which illustrates the case f ( x ) ≥ 0.
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Chap7_Sec7 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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