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

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MATH101 April 2009

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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).

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

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

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 ${\begin{aligned}S_{n}&={\frac {1}{2\cdot 3}}\left(f(0)+4f(0.5)+2f(1)+4f(1.5)+f(2)\right)\\&={\frac {1}{6}}(\sin(e^{0})+4\sin(e^{0.5})+2\sin(e^{1})+4\sin(e^{1.5})+\sin(e^{2}))\end{aligned}}$

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