Question

Question 1: Numerically integrate function f(x) given on the right from x=0 to x=10. Use the f(x)= x2 - 6x + 16 Trapezoidal Rule and the Simpson's 1/3 Rule and compare the results. Use at least 4 50 + 15x - ye? decimal digits in your calculations and reporting. Organize and report each one of your solutions in a calculation table and identify your result clearly. a) Divide the interval into 5 subdivisions. Calculate the integral first using the Trapezoidal Rule and then the Simpson's 1/3 Rule and compare the results. b) Divide the interval into 10 subdivisions. Calculate the integral first using the Trapezoidal Rule and then the Simpson's 1/3 Rule and compare the results.

Answer 1

$\int_{a}^{b}f(x)dx:\textbf{Trapezoidal Rule}$

Tn = AC f(20) 2 + f(21) + f(22) + ... + f(2n-1) + 2

b - a where to = a, In = b, A.r = n

$\int_{a}^{b}f(x)dx:\mathbf{Simpson's~\frac{1}{3}~ Rule}~~(n\textup{ must be even)}$

Sn (f(20) + 4f(21) +2f(12) + 4f(13) + ... +2f(In-2) + 4f(In-1) + f(In) 3

b - a where to = a, In = b, A.r = n

$(a)~\textup{Question is wrong, Simpson's rule cannot be applied for }n=5$

$f(x)=\frac{x^{2}-6x+16}{50+15x-x^{2}}$

$n=5:~x_{0}=0,~~x_{5}=10,~~\Delta x=\frac{10-0}{5}=2$

$T_{5}=2\left \{ \frac{f(x_{0})}{2}+f(x_{1})+f(x_{2})+f(x_{3})+f(x_{4})+\frac{f(x_{5})}{2} \right \}$

NOX 4 Trapezoidal Rule: n = 5 f(x) Trapezoidal Sum 0.32 0.32 0.105263158 0.210526316 0.085106383 0.170212766 0.153846154 0.307692308 0.301886792 0.603773585 0.56 0.56 2.172204974 6 8 10

$T_{5}\approx 2.1722$

$(b)~n=10:~ x_{0}=0,~~x_{5}=10,~~\Delta x=\frac{10-0}{10}=1$

$T_{10}= \frac{f(x_{0})}{2}+f(x_{1})+f(x_{2})+...+f(x_{9})+\frac{f(x_{10})}{2}$

XO 1 N 3 4 Trapezoidal Rule: n = 10 f(x) Trapezoidal Sum 0.32 0.16 0.171875 0.171875 0.105263158 0.105263158 0.081395349 0.081395349 0.085106383 0.085106383 0.11 0.11 0.153846154 0.153846154 0.216981132 0.216981132 0.301886792 0.301886792 0.413461538 0.413461538 0.56 0.28 5 6 7 8 9 10 2.079815507

$T_{10}\approx 2.0798$

$S_{10}=\frac{1}{3}\left ( f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+...+2f(x_{8})+4f(x_{9})+f(x_{10}) \right )$

X 0 1 N 3 4 Simpson's Rule: n = 10 f(x) Trapezoidal Sum 0.32 0.106666667 0.171875 0.229166667 0.105263158 0.070175439 0.081395349 0.108527132 0.085106383 0.056737589 0.11 0.146666667 0.153846154 0.102564103 0.216981132 0.289308176 0.301886792 0.201257862 0.413461538 0.551282051 0.56 0.186666667 2.049019017 5 6 7 8 9 10

$S_{10}\approx 2.049$

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