Science:Math Exam Resources/Courses/MATH103/April 2014/Question 04 (a)
• Q1 (a) • Q1 (b) i • Q1 (b) ii • Q1 (c) i • Q1 (c) ii • Q1 (c) iii • Q1 (d) i • Q1 (d) ii • Q1 (d) iii • Q1 (e) i • Q1 (e) ii • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (c) • Q7 (a) • Q7 (b) • Q7 (c) • Q8 (a) • Q8 (b) • Q9 (a) • Q9 (b) • Q9 (c) • Q10 (a) • Q10 (b) • Q10 (c) • Q11 •
Question 04 (a) 

Evaluate if it exists, or show that it does not exist, otherwise. Work must be shown for full marks. Simplify fully. 
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 

Note that the sum is over (not ). Thus, is a constant relative to the sum. 
Hint 2 

Recall the definition of the integral as a limit of Riemann sums. 
Hint 3 

The Riemann sum formula (for right endpoints) is where . Compare this formula to what is given to determine the unknowns. 
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.

Solution 1 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. By definition of the integral (see Hint 3), Comparing to the original sum we have that . Therefore, 
Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. There is an alternative solution if you remember the sum of squares:
With this we quickly find
