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

Use a Riemann sum with n = 2 and select the midpoints of subintervals to estimate the area under the graph of from 0 to 4. 
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 

How many intervals will you have to use? Where will their endpoints be? What about their midpoints? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. With n = 2 and an interval of we have and i.e. our subinterval endpoints are 0, 2 and 4. To approximate with midpoints, we need to take sample points (these are the midpoints of and , and and respectively). The area under the curve is approximated by the midpoint Riemann sum, which yields 