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$(a)$ Sketch the region of integration in the integral

$$\int_{y=-2}^{2} \int_{x=0}^{\sqrt{4-y^2}} x e^{{(4-x^{2})}^{3/2}} dx dy$$

By changing the order of integration, or otherwise, evaluate the integral.

$(b)$ Let $R$ be the region in the $x-y$ plane defined by $0 \leq x \leq y \leq 2x$, $1 \leq x+2y \leq 4$. Evaluate: $$\mathop{\int\int}_{R} \frac{1}{x} dx dy$$

I understand how to draw these but I am not sure how to caluculate the limts in either case (especially part $b$).

Can someone explain how we calculate the limits for integration? Once I know that I am sure I can integrate the function myself. Thanks!!

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    You are asking too many homework questions. It won't particularly benefit you to get any particular question answered because you do not seem to have a strong grasp of the underlying material, so I would suggest that you go to a professor or TA and/or reread your textbook more thoroughly and then think harder about these questions on your own. Constantly tossing out homework questions is not the purpose of this site.2011-05-21
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    This is not a homeowrk question. Im preparing for an exam and understand the steps but Ive never had a quadrilateral area to work out. The only examples we were given were triangles. Therefore I still dont know how to find the limits.2011-05-23

2 Answers 2

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The first thing, as stated, is to sketch the region. Only then calculate the limits of integration. So, what does your sketch look like?

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    +1: but it might be hard to show his/her sketch.2011-05-21
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    For part (a). Its a circle going through (0,2), (2,0), (-2,0) and (0,-2). So we are integrating that top half of the circle. But what do we take for x and y? Why do we switch the order? And part (B)'s limits make even less sense to me! :(2011-05-21
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    For part (a), [this](http://www.wolframalpha.com/input/?i=0%E2%89%A4x%E2%89%A4y%E2%89%A42x+%26%26+1%E2%89%A4x%2B2y%E2%89%A44) is sketch of the region.2011-05-21
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    Yep its not the sketch thats the problem. Its the limits. What do we fix? x or y? what values? How do we calulate the x limits? Thats what confusing me. Can you explain it in general what o do so that I know for other questions? Thanks.2011-05-21
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    You fix something (either $x$ or $y$), this depends on the order of integration you doing. Then you determine over which range the other variable has to go (obviously depending on the value of the first value which you have chosen) to cover the whole area.2011-05-21
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    @user4645: First identify and sketch carefully your region. Then suppose you are integrating first with respect to $y$. Then $y$ goes from "bottom curve" to top curve." So if bottom curve is $y=f(x)$ and top curve is $y=g(x)$, integrate with respect to $y$ from $f(x)$ to $g(x)$. Then integrate with respect to $x$ from the first $x$ to the last. Sometimes, as in (b), top and bottom curves change. Then break up your region into parts where they don't change. In (b), $3$ parts will do the job. You really need someone at a blackboard pointing, or a video. This stuff is very visual.2011-05-21
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    @user6312: Just tried it: Click on the offending long line, and highlight it towards the right, this will make the comment scroll until the delete (and the edit) button appear.2011-05-22
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    Hi. I have sketched it as well as ossible and got the correct answer - its the same as Wolframs. Im still having major problems picking he correct limits. Can you please explain how to split this up and how to pick limits? I really dont understand it at all. We have only been taught how to integrate triangles and therefore this makes no sense to me... I have gone to lectures and stuff but no examples are given hence my serious confusing. Btw this is not homeowkr - Im preparing for a exam thats in 1 week!2011-05-22
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    @user9325: Thanks, I hope I now know how to do it. Unfortunately, the problem had already been fixed, so I did not get to do it. Good metaphor maybe for what we do in solving problems for students.2011-05-22
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    Ive tried again and again but i really dont know what limits to take. Can someone please explain this example to me? Ive never had a quadrilateral area before. All the examples given in doube integration are triangles (which i get)... Please... Im seriously stuck on this and until I dont get this I cant do any similar question. :( I have looked in books/lecture notes/asked other people but the information is unclear.2011-05-23
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    PLEASE PLEASE PLEASE someone help me. im seriously failing all of tese questions because Im clueless how to find the limits.2011-05-24
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    Hmmm... asked 19 questions but never accepted a single answer ... why should we help you?2011-05-29
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    sorry i have corrected my mistakes!2011-05-30
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  1. The integral is taken over the upper semicircle, so $$\int\limits_{-2}^2\:dy\int\limits_0^\sqrt{4-y^2}xe^{(4-x^2)^\frac32}\:dx = \int\limits_0^2xe^{(4-x^2)^\frac32}\:dx\int\limits_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\:dy = 2\int\limits_0^2x\sqrt{4-x^2}e^{(4-x^2)^\frac32}\:dx =$$$$ -\dfrac23\int\limits_0^2e^{(4-x^2)^\frac32}\left((4-x^2)^\frac32\right)'\:dx = -\left.e^{(4-x^2)^\frac32}\right|_0^2 = e^8-1$$