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Approximate Integration

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Approximate Integration

1. 1. Approximate Integration T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
2. 2. Integration Techniques  There are two situations in which it is impossible to find the exact value of a definite integral.  The first situation arises from the fact that, in order to evaluate using the Fundamental Theorem of Calculus (FTC), we need to know an anti-derivative of f.  However, sometimes, it is difficult, or even impossible, to find an anti-derivative.  For example, it is impossible to evaluate the following integrals exactly:  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 : 21 1 3 0 1 1x e dx x dx ( ) b a f x dx © iTutor. 2000-2013. All Rights Reserved
3. 3. 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.  If we divide [a, b] into n subintervals of equal length ∆x = (b – a)/n, we have: where xi* is any point in the i th subinterval [xi -1, xi]. Ln Approximation  If xi* is chosen to be the left endpoint of the interval, then xi* = xi -1 and we have:  The approximation Ln is called the left endpoint approximation. 1 ( ) ( *) nb ia i f x dx f x x 1 1 ( ) ( ) nb n ia i f x dx L f x x ------- Equation 1 © iTutor. 2000-2013. All Rights Reserved
4. 4.  If f(x) ≥ 0, the integral represents an area and Equation 1 represents an approximation of this area by the rectangles shown here. Rn Approximation  If we choose xi* to be the right endpoint, xi* = xi and we have:  The approximation Rn is called right endpoint approximation. 1 ( ) ( ) nb n ia i f x dx R f x x © iTutor. 2000-2013. All Rights Reserved
5. 5.  The figure shows the midpoint approximation Mn.  Mn appears to be better than either Ln or Rn. Mn Approximation © iTutor. 2000-2013. All Rights Reserved
6. 6. Midpoint Rule  Let f be continuous on [a, b].  The Midpoint Rule for approximating is given by x y a 1 x 2 x bnx… )(xfy b a dxxf )( 1 2 ( ) [ ( ) ( ) ... ( )] b na n f x dx M x f x f x f x b a x n 1 1 12 ( ) midpoint of [ , ]i i i i ix x x x x Where and
7. 7. .72135.3235.2235.12 333 EXAMPLE SOLUTION Approximate the following integral by the midpoint rule. We have Δx = (b – a)/n = (4 – 1)/3 = 1. The endpoints of the four subintervals begin at a = 1 and are spaced 1 unit apart. The first midpoint is at a + Δx/2 = 1.5. The midpoints are also spaced 1 unit apart. According to the midpoint rule, the integral is approximately equal to 3;32 4 1 3 ndxx © iTutor. 2000-2013. All Rights Reserved
8. 8.  Let f be continuous on [a, b]. The Trapezoidal Rule for approximating is given by Trapezoidal Rule b a dxxf )( 0 1 2 1 ( ) ( ) 2 ( ) 2 ( ) 2 ... 2 ( ) ( ) b na n n f x dx T x f x f x f x f x f x where ∆x = (b – a)/n and xi = a + i ∆x © iTutor. 2000-2013. All Rights Reserved
9. 9.  The reason for the name can be seen from the figure, which illustrates the case f(x) ≥ 0.  The area of the trapezoid that lies above the i th subinterval is:  If we add the areas of all these trapezoids, we get the right side of the Trapezoidal Rule. 1 1 ( ) ( ) [ ( ) ( )] 2 2 i i i i f x f x x x f x f x © iTutor. 2000-2013. All Rights Reserved
10. 10. .90 2 1 34233223222312 3333 EXAMPLE SOLUTION Approximate the following integral by the trapezoidal rule. As in the last example, Δx = 1 and the endpoints of the subintervals are a0 = 1, a1 = 2, a2 = 3, and a3 = 4. The trapezoidal rule gives 3;32 4 1 3 ndxx © iTutor. 2000-2013. All Rights Reserved
11. 11. Simpson’s Rule  Rather than using straight lines to approximate the curve, Simpson’s Rule uses parabolas. 0 1 2 3 2 1 ( ) [ ( ) 4 ( ) 2 ( ) 4 ( ) 3 ... 2 ( ) 4 ( ) ( )] b na n n n f x dx S x f x f x f x f x f x f x f x Where n is even and ∆x = (b – a)/n. © iTutor. 2000-2013. All Rights Reserved
12. 12. EXAMPLE SOLUTION Putting f(x) = 1/x, n = 10, and ∆x = 0.1 in Simpson’s Rule, we obtain: Use Simpson’s Rule with n = 10 to approximate 2 1 (1/ )x dx 2 101 1 [ (1) 4 (1.1) 2 (1.2) 4 (1.3) 3 ... 2 (1.8) 4 (1.9) (2)] 1 4 2 4 2 4 2 4 0.1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2 4 13 1.8 1.9 2 0.693150 x dx S f f f f x f f f © iTutor. 2000-2013. All Rights Reserved
13. 13. The End Call us for more Information: www.iTutor.com 1-855-694-8886 Visit