Science:Math Exam Resources/Courses/MATH105/April 2016/Question 01 (e)

MATH105 April 2016

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q1 (h)  •  Q1 (i)  •  Q1 (j)  •  Q1 (k)  •  Q1 (l)  •  Q1 (m)  •  Q1 (n)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q3  •  Q4 (a)  •  Q4 (b)  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q5 (d)  •  Q6  •  

   Other MATH105 Exams

  •  April 2009  •  April 2011  •  April 2010  •  April 2012  •  April 2013  •  April 2014  •  April 2015  •  April 2016  •  April 2017  •  April 2018  •  

Question 01 (e)
Does a left Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and decreasing on an interval $[a,b]$?

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Definition of left Riemann sum

Hint 2
Sketch any function which is positive and decreasing on [a,b], and compare the area of the left Riemann sum with the area under the graph of the function you sketched.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. First, since the function is positive, the area of the region under the graph of the function is ${\color {OrangeRed}\int _{a}^{b}f(x)dx}$. On the other hand, the left Riemman sum of the area $n$ subintervals is given by ${\color {OliveGreen}\sum _{i=1}^{n}f(a+(i-1)\Delta x)\Delta x}$, where $\Delta x={\frac {b-a}{n}}.$ Since the function is a decreasing, on each subinterval , the function value of left endpoint is greater than the function value of any other points on the subinterval; In other words , for each $x$ in ith subinterval, (i.e., $x\in [a+(i-1)\Delta x,a+i\Delta x]$), we have $f(a+(i-1)\Delta x)\geq f(x)$. This implies that ${\color {OrangeRed}\int _{a}^{b}f(x)dx}=\sum _{i=1}^{n}\int _{a+(i-1)\Delta x}^{a+i\Delta x}f(x)dx\quad {\color {blue}\leq }\quad \sum _{i=1}^{n}\int _{a+(i-1)\Delta x}^{a+i\Delta x}f(a+(i-1)\Delta x)dx=\sum _{i=1}^{n}f(a+(i-1)\Delta x)\int _{a+(i-1)\Delta x}^{a+i\Delta x}dx={\color {OliveGreen}\sum _{i=1}^{n}f(a+(i-1)\Delta x)\Delta x}.$ Thus, the left Riemann sum overestimates the area.