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

MATH105 April 2010

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[hide] 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.

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

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[show] Solution
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 ${\begin{aligned}\int _{0}^{4}f(x)\,dx&\approx \Delta x(f(x_{1}^{})+f(x_{2}^{}))\\&=2\left({\frac {1}{1+{\sqrt {1}}}}+{\frac {1}{1+{\sqrt {3}}}}\right)\\&=1+{\frac {2}{1+{\sqrt {3}}}}\end{aligned}}$