Science:Math Exam Resources/Courses/MATH105/April 2010/Question 01 (f)

MATH105 April 2010

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Question 01 (f)
Use a Riemann sum with n = 2 and select the midpoints of subintervals to estimate the area under the graph of $f(x)={\frac {1}{1+{\sqrt {x}}}}$ from 0 to 4.

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
How many intervals will you have to use? Where will their endpoints be? What about their midpoints?

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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. With n = 2 and an interval of $[0,4]$ we have $\Delta x={\frac {4-0}{2}}=2$ and $x_{j}=0+j\Delta x=2j\quad {\text{for }}j=0,1,2$ i.e. our subinterval endpoints are 0, 2 and 4. To approximate with midpoints, we need to take sample points $x_{1}^{*}=1\quad {\text{and}}\quad x_{2}^{*}=3$ (these are the midpoints of $x_{0}$ and $x_{1}$, and $x_{1}$ and $x_{2}$ respectively). The area under the curve is approximated by the midpoint Riemann sum, which yields ${\begin{aligned}\int _{0}^{4}f(x)\,dx&\approx \Delta x(f(x_{1}^{*})+f(x_{2}^{*}))\\&=2\left({\frac {1}{1+{\sqrt {1}}}}+{\frac {1}{1+{\sqrt {3}}}}\right)\\&=1+{\frac {2}{1+{\sqrt {3}}}}\end{aligned}}$

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