Science:Math Exam Resources/Courses/MATH101/April 2012/Question 07 (a)

MATH101 April 2012

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Question 07 (a)
Let $I=\int _{1}^{2}(1/x)\,dx$. Write down the trapezoidal approximation T4 for I. You do not need to 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 general formula for the trapezoidal approximation $T_{n}$ is given by: $T_{n}={\frac {\Delta x}{2}}(f(x_{0})+2f(x_{1})+\dots +2f(x_{n-1})+f(x_{n}))$ What are $x_{0}$ and $x_{n}$?

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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 trapezoidal approximation of the integral is as follows: $T_{4}={\frac {\Delta x}{2}}(f(x_{0})+2f(x_{1})+2f(x_{2})+2f(x_{3})+f(x_{4}))$ The 4 of $T_{4}$ means that our interval of integration has been divided into four pieces, each of length (b-a)/4=1/4. So $\Delta x=1/4$. We also know that $x_{0}$ and $x_{4}$ are the endpoints of the interval of integration. So we have ${\begin{aligned}x_{0}&=1\\x_{1}&=5/4\\x_{2}&=6/4\\x_{3}&=7/4\\x_{4}&=2\\\end{aligned}}$ We simply plug this all into the formula given above to get: $T_{4}={\frac {1}{8}}\left({\frac {1}{1}}+2{\frac {1}{5/4}}+2{\frac {1}{6/4}}+2{\frac {1}{7/4}}+{\frac {1}{2}})\right)$

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