Science:Math Exam Resources/Courses/MATH105/April 2016/Question 02 (b)

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MATH105 April 2016

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Question 02 (b)
Give the right Riemann sum for $f(x)=x^{\frac {3}{2}}$ on the interval $[2,6]$ with n subintervals of equal length.

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
Definition of right Riemann sum.

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Solution
The right Riemann sum for function $f(x)$ on the interval $[a,b]$ with $n$ subintervals of equal length is given by $\sum _{i=1}^{n}f(x_{i})\Delta x,$ where $x_{i}=a+i\Delta x$ and $\Delta x={\frac {b-a}{n}}.$ Here $f(x)=x^{\frac {3}{2}},$ $a=2$ and $b=6.$ So, $\Delta x={\frac {6-2}{n}}={\frac {4}{n}}.$ Also, for $i=1,2,3,...,n$ , we have $x_{i}=2+i\Delta x=2+{\frac {4i}{n}},$ Thus, the Riemann sum is $\sum _{i=1}^{n}f(x_{i})\Delta x={\color {Blue}\sum _{i=1}^{n}(2+{\frac {4i}{n}})^{\frac {3}{2}}{\frac {4}{n}}}$