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

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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 . On the other hand, the left Riemman sum of the area subintervals is given by , where 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 in ith subinterval, (i.e., ), we have
This implies that
Thus, the left Riemann sum overestimates the area. 