MTH405_Wk_10_Lect_2 - MTH405 Wk 10 Lect 2 Numerical...

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Unformatted text preview: MTH405 Wk 10 Lect 2 Numerical Integration Numerical Integration is a also another way of nding area under the graph, but it is very grounded. Most of the time, Numerical Integration is used to nd area for functions which can not be integrated (Yes, there are functions for which the techniques of integration we have learnt can not be applied). Most importantly, notice that when we started Integration topic, we stated with 100% condence that the Area under the Graph is given by ˆ b f (x)dx. A= a But we did not show how this is true. (We are still trying to answer the AREA PROBLEM) This is true because the actual formula for nding Area under the Graph is given by A = lim n→+∞ n X f (x∗k )∆x. k=1 This is called the Riemann Sum. And there is a theorem that states that ˆ n b f (x)dx = lim a n→+∞ 1 X k=1 f (x∗k ) · ∆x. Summary on Riemann Sum We need to nd the area under the curve along the interval [a, b]. We use Archimedes method of exhaustion by dividing the interval into n equal subintervals as shown below. This method is known as the Rectangle method as we have many slim rectangles in each subinterval. Then the width of each subinterval is given by ∆x = b−a . n Hence, the endpoints of the subintervals are listed as a = x0 , x1 , x2 , . . . , xk , . . . , xn = b. To give us the the height of the rectangles, we take a point x from each subinterval: x∗1 , x∗2 , . . . , x∗k , . . . , x∗n−1 , x∗n , Hence the heights of the rectangles are f (x∗1 ), f (x∗2 ), . . . , f (x∗k ), . . . , f (x∗n−1 ), f (x∗n ). 2 The area for each rectangle is given by Ak = H × W = f (x∗k ) · ∆x. Then taking the sum of all rectangles gives the area A≈ n X f (x∗k ) · ∆x. k=1 This is the APPROXIMATION FORMULA. 3 But that is just an approximation; to get a more accurate value of area, we increase the number of subintervals n → +∞ (the more the subintervals n, the smaller the rectangles and lesser the error). Since we can't just say n = +∞, then we just take it to the limit: A = lim n→∞ n X f (x∗k ) · ∆x. k=1 This is the rigorous AREA FORMULA for area under the graph, but this won't be covered in this course. Midpoint Approximation Note the approximation formula given before A≈ n X f (x∗k ) · ∆x. k=1 This is the approximation we will be using for Midpoint, Trapezoidal and Simpson approximations. For Midpoint approximation, we have the following formulae to use: width of the subintervals, ∆x = b−a n 1 2 choice of midpoints of the subintervals, x∗k = a + (k − )∆x Area function, A = ˆ b  f (x)dx ≈ a Example. Evaluate Example. Evaluate ´2 1 dx 1 x ´3 1 b−a n  [f (x∗1 ) + f (x∗2 ) + . . . + f (x∗n )] where n = 4 using Midpoint Rule. (x3 + 1)dx where n = 4 using Midpoint Rule. 4...
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