METBD 050 – Trapezoid Method.doc Page 1 of 3 1/29/02, Rev. 3 – 10/1/03 METBD 050 Trapezoid Method: The trapezoid method is a method for finding the area under the graph of a function f(x) between two values of x. While there are several other methods available to do this, we will discuss the simplest method which is known as the trapezoid method. Using the trapezoid method, we divide the area into a number of equally wide trapezoids, one of which is shown in the figure below. The sum of all of the trapezoids is the approximate area under the graph. As the number of trapezoids increases, the panel width, ∆X, gets smaller, and the result approaches the exact solution. The area under the curve is given by: The area of the strip is: 2YYXA1iii)(++∆=. Combining this expression with the definition of the area above, we have: )()()()(1nn2ii11nni3211nn1ii3221YY2Y2XAYY2Y2Y2Y2Y2XA2YYX2YYX2YYX2YYXA+=+++++∆≈+++++∆≈+∆+∆++∆++∆≈∑KKKK In the above expressions, ∆X = the width of each panel and n is the number of panels used.f(x) X = a Xi Xi+1X = b Yi Yi+1∆XAiY X ∑∫=≅=n1iibaAdxxfA )(TRAPEZOID RULE TrapezoidMETBD 050 – Trapezoid Method.doc Page 2 of 3 1/29/02, Rev. 3 – 10/1/03 EXAMPLE: Find the area under the curve y = -x2 + 9 between x = 0 and x = 3. In the solution of this problem, we will use three panel widths, therefore, n = 3. ∆X = 3 / 3 panels = 1. 517A0582921AYY2Y2XAn1i1ni1.))(()(≈+++≈++∆≈∑=+ Incidently the exact area is 189332bh32A === ))((. The approximate solution has an error of 2.78%. By increasing the number of panels, we can increase the accuracy of the approximate solution. The flowchart: Note: For those interested, the calculus solution for area is: (0, 9) (1, 8) (2, 5) 3 2 1 0 y = -x2 + 9 (3, 0) Start Get a, b, & n Dx = (b-a)/n ya = -(a^2)+9 yb = -(b^2)+9 x = x + Dx Sum = 0 Is x < b?YESyi = -(x^2)+9 Sum = Sum + yi x = x + Dx NOA = .5*Dx*(ya+2*Sum+yb) Display A End ∫=+−=30218dx9xA )(METBD 050 – Trapezoid Method.doc Page 3 of 3 1/29/02, Rev. 3 – 10/1/03 The following worksheet implements the algorithm given above. Note that the problem is actually solved two times – once using three panel widths, and once using 6 panel widths. Also note the improvement in accuracy by doubling the number of panels. ABCDEFGH I1Worksheets finds area under y = -x2 + 9 between x = 0 and 3.23Trial 1: 3 panels Trial 2: 6 panels45 a = 0 = start point a = 0 = start point6 b = 3 = end point b = 3 = end point7 n = 3 = number of panels n = 6 = number of panels89 Dx = 1 Dx = 0.51011 y y12 ya = 9 ya = 913 yb = 0 yb = 01415xy xy16 1 8.00 0.5 8.7517 2 5.00 1.0 8.0018 Sum = 13.00 1.5 6.7519 2*Sum = 26.00 2.0 5.0020 2.5 2.7521 Area = 17.5Sum = 31.2522 %Error = 2.78 2*Sum = 62.502324 Area = 17.87525 %Error = 0.692627 Cell Formulas Used for Trial 1: Cell Cell Formula B9 =(B6-B5)/B7 B12 =-(B5^2)+9 B13 =-(B6^2)+9 A16 =B5+B9 B16 =-(A16^2)+9 A17 =A16+$B$9 B17 =-(A17^2)+9 B18 =SUM(B16:B17) B19 =2*B18 B21 =.5*B9*(B12+B19+B13) C21
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