Science:Math Exam Resources/Courses/MATH200/April 2012/Question 06 (a)

MATH200 April 2012

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  •  April 2012  •  December 2011  •  

Question 06 (a)
Combine the sum of the iterated integrals $I=\int _{0}^{1}\int _{-{\sqrt {y}}}^{\sqrt {y}}f(x,y){\textrm {d}}x{\textrm {d}}y+\int _{1}^{4}\int _{y-2}^{\sqrt {y}}f(x,y){\textrm {d}}x{\textrm {d}}y$ into a single iterated integral with the order of integration reversed.

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
Try making a sketch of the region of integration first, rather than trying to tackle this problem from a purely algebraic standpoint.

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. 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. If we examine the region of integration by drawing a picture... The domain of integration is enclosed within the curves $\displaystyle x=y-2$ (black), ${\color {blue}x=-{\sqrt {y}}}$ (blue), and ${\color {red}x={\sqrt {y}}}$ (red) ... we see that if we integrate with respect to y first, that we can specify the lower bound of integration as $y=x^{2}$ and the upper bound as $y=x+2$. Looking at where the lines meet, we can see that the bounds of integration for x is x = -1 to x = 2. (To find these intersection points solve the equation $x^{2}=x+2$.) From these observations, we can write I as follows: ${\color {blue}I=\int _{-1}^{2}\int _{x^{2}}^{x+2}f(x,y)\,{\textrm {d}}y{\textrm {d}}x}$