Science:Math Exam Resources/Courses/MATH101/April 2006/Question 02 (a)
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Question 02 (a) 

FullSolution Problems. Justify your answers and show all your work. Simplification of answers is not required. Sketch the bounded region that lies between the curves and , and show that the area of this region equals . 
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 

Find the points of intersection  then draw the picture. 
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. A picture is included below. First, find the points of intersection.
Take the positive values of first and solving gives
and so is a solution here (notice that we discard since we started with only the positive x values first). Now, taking the negative x values gives
and so is the other solution. Now, we can draw the picture (see below). The picture clearly shows that the upper function is 2x^{2}, and that the function is an even function. So we see that it suffices to compute
Evaluating gives
as required. 