and this is an example Set Up The objective of the project is to confirm empiric

Question

and this is an example

Set Up The objective of the project is to confirm empirically the veracity of integral calculus. That is, we will find the area under an arbitrary curve using technology and compare it with what we find using "anti-differentiation". Leibnitz felt that the connection of finding area to differential calculus proved that God exists; we will "prove" that the connection exists! Page One is a suitably impressive cover page and is worth ten points Page Two requires primarily two tables produced using Excel. For the first table, choose a polynomial of degree 3 or higher, or a radical expression such as Vx; as well as an interval over which the area under this curve is to be computed. Divide this interval into at least 40 sub- intervals of fractional size (less than or equal to 12) and proceed to approximate area using Upper-Rectangles, the Trapezoidal Rule, and Simpson's Rule as explained in class. Compare your solutions with the exact answer found by integration YSur second table will be identical to your first table except here you will find the area under a curve where integration is not possible, the well-known Bell-Shaped curve used in statistics: y ex More precisely, you are to verify that the area from 0 to oo is 1.2533136 which approximates ½d2n). (*The actual Bell-Curve is y = 1/[V(2n)] . e-x22 as the total area beneath it from -co to co must be 1, a necessity of any curve used in probability theory.) Cooperation among students is encouraged but no two projects are to be identical Example: y-Vx over [0,3] sub-intervals to be h-½ in length Rectangle Rle Trapezoidal Rule Simpson's Rule 0 Sum RHS Sum cells AI &A2Sum; Al & 4 A2 & A3 707 707 707 3.828 .299 1.414 1.581 1.732 1.414 .581 1.732 2.995 3.313 Rectangle Rule: h., column values ½(7.658) -3.829 Trapezoidal Rule: h/2 . column values- 14(13.877) 3.469 Simpson's Rule: his . y column values- 16 (19.959) 3.325 By integration fokia" 2/3yWb3 3.429

Explanation / Answer

So, Simpson's rule and Trapezoidal rule are very accurate

y=e^(-x^2/2) x y Rectangle rule Trapezoidal rule Simpson's rule 0 1 1 0.5 0.882496903 0.882496903 1.882496903 1 0.60653066 0.60653066 1.489027562 5.136518 1.5 0.324652467 0.324652467 0.931183127 2 0.135335283 0.135335283 0.459987751 2.040476 2.5 0.043936934 0.043936934 0.179272217 3 0.011108997 0.011108997 0.05504593 0.322192 3.5 0.002187491 0.002187491 0.013296488 4 0.000335463 0.000335463 0.002522954 0.020194 4.5 4.00653E-05 4.00653E-05 0.000375528 5 3.72665E-06 3.72665E-06 4.3792E-05 0.000499 5.5 2.69958E-07 2.69958E-07 3.99661E-06 6 1.523E-08 1.523E-08 2.85188E-07 4.82E-06 6.5 6.69159E-10 6.69159E-10 1.58991E-08 7 2.28973E-11 2.28973E-11 6.92056E-10 1.79E-08 7.5 6.10194E-13 6.10194E-13 2.35075E-11 8 1.26642E-14 1.26642E-14 6.22858E-13 2.54E-11 8.5 2.04697E-16 2.04697E-16 1.28689E-14 9 2.57676E-18 2.57676E-18 2.07274E-16 1.35E-14 9.5 2.52616E-20 2.52616E-20 2.60202E-18 10 1.92875E-22 1.92875E-22 2.54545E-20 2.68E-18 10.5 1.14688E-24 1.14688E-24 1.94022E-22 11 5.31109E-27 5.31109E-27 1.15219E-24 1.97E-22 11.5 1.91548E-29 1.91548E-29 5.33025E-27 12 5.38019E-32 5.38019E-32 1.92086E-29 5.39E-27 12.5 1.17691E-34 1.17691E-34 5.39196E-32 13 2.00501E-37 2.00501E-37 1.17892E-34 5.43E-32 13.5 2.66021E-40 2.66021E-40 2.00767E-37 14 2.74879E-43 2.74879E-43 2.66296E-40 2.02E-37 14.5 2.21204E-46 2.21204E-46 2.751E-43 15 1.38634E-49 1.38634E-49 2.21342E-46 2.76E-43 15.5 6.76668E-53 6.76668E-53 1.38702E-49 16 2.57221E-56 2.57221E-56 6.76925E-53 1.39E-49 16.5 7.61489E-60 7.61489E-60 2.57297E-56 17 1.75569E-63 1.75569E-63 7.61665E-60 2.58E-56 17.5 3.15252E-67 3.15252E-67 1.756E-63 18 4.40853E-71 4.40853E-71 3.15296E-67 1.76E-63 18.5 4.80128E-75 4.80128E-75 4.40901E-71 19 4.07236E-79 4.07236E-79 4.80168E-75 4.41E-71 19.5 2.69006E-83 2.69006E-83 4.07263E-79 20 1.3839E-87 1.3839E-87 2.6902E-83 4.07E-79 Total 3.006628275 3.006628275 5.013256549 7.519885 Applied 1.503314137 1.253314137 1.253314

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