Science:Math Exam Resources/Courses/MATH105/April 2018/Question 01 (e)

MATH105 April 2018

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Question 01 (e)
(e) Use sigma notation to write the right Riemann sum for ${\textstyle f(x)=x+2}$ on ${\textstyle [0,6]}$ with ${\textstyle n=60}$. DO NOT EVALUATE the right Riemann sum.

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 right Riemann sum for ${\textstyle f(x)}$ with the partition ${\textstyle P={\big \{}[x_{0},x_{1}],[x_{1},x_{2}],\dots ,[x_{n-1},x_{n}]{\big \}}}$ is given by $$\sum _{i=1}^{n}f(x_{i})(x_{i}-x_{i-1}).$$

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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. Since $n=60$ and the given interval for the Riemann sum is $[0,6]$. This implies that each interval in the partition has the size ${\frac {1}{10}}={\frac {6}{60}}$, so we will use the partition ${\textstyle P={\big \{}[0,1/10],[1/10,2/10],\dots ,[6-1/10,6]{\big \}}}$. Then plugging ${\textstyle f(x)=x+2}$ and ${\textstyle n=60}$ into the formula in the hint we see that the right Riemann sum is precisely $$\sum _{i=1}^{60}{\frac {f(i/10)}{10}}=\sum _{i=1}^{60}\left({\frac {i}{10}}+2\right){\frac {1}{10}}.$$ Answer: The right Riemann sum is given by $\color {blue}\sum _{i=1}^{60}\left({\frac {i}{10}}+2\right){\frac {1}{10}}.$