Science:Math Exam Resources/Courses/MATH101/April 2009/Question 01 (j)

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Question 01 (j)
Give the Simpson’s rule approximation to $\int _{0}^{2}\sin(e^{x})\,dx$ using 4 equal subintervals (do not simplify your answer).

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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The formula for the Simpson's rule is given by $S_{n}:={\frac {\Delta x}{3}}\left(f(x_{0})+4f(x_{1})+2f(x_{2})+...+4f(x_{n-1})+f(x_{n})\right)$ where as always $n$ is an even integer.

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The formula for the Simpson's rule is given by $S_{n}={\frac {\Delta x}{3}}\left(f(x_{0})+4f(x_{1})+2f(x_{2})+...+4f(x_{n-1})+f(x_{n})\right)$ When $n=4$ , we have that $x_{i}=a+i\Delta x=0+{\frac {(2-0)i}{4}}={\frac {i}{2}}$ and thus, the Simpson's rule approximation that we need is ${\begin{aligned}S_{n}&={\frac {1}{2\cdot 3}}\left(f(0)+4f(0.5)+2f(1)+4f(1.5)+f(2)\right)\\&={\frac {1}{6}}(\sin(e^{0})+4\sin(e^{0.5})+2\sin(e^{1})+4\sin(e^{1.5})+\sin(e^{2}))\end{aligned}}$