Basically, like I said in a topic a while ago, I'm taking Calculus III under circumstances which I have to basically teach myself from the book, and I've run into a bit of a wall, and I'd greatly appreciate any help.
The book is limiting dealing with nonrectangular regions of type I and type II. I'm going through this set of problems where I'm given a region, then supposed to evaluate it first as a type I, then as a type II. I've gotten a correct answer after setting it up as a type I region, but I'm running into a problem trying to evaluate it as a type II region.
Basically, the problem amounts to this: I'm supposed to find the double integral of x^2 over the region bounded by y=16/x, y=x, and y=8. Here's how I set up the Type I region:
http://img207.imageshack.us/img207/1958/calctype1.png
This setup worked just fine, giving me an answer of 576.
Then, I tried to set it up as a type II region as follows:
http://img209.imageshack.us/img209/3531/calctype2.png
Unfortunately, this gave me a different answer of 192. This leads me to believe that I've either incorrectly set up the limits of integration, or that I've made an error in the integration. So basically I'd like to know if I've set up the limits of integration wrong like I suspect, and if I have, where did I go wrong. To help assist that, for the type II region, I made a diagram to show where I pulled the limits from:
http://img529.imageshack.us/img529/8522/calcdiag.png
Many thanks in advance!
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I thought I saw upon the stair a little man who wasn't there.
He wasn't there again today. Oh how I wish he'd go away.
Looks like you merely made a mistake evaluating calctype2.png, since my computer algebra system (CAS) says it's equal to 576. If you're not already using a CAS, I recommend Maxima:
http://maxima.sourceforge.net/
Maxima is the program I used to evaluate these integrals. Yes, you're supposed to be able to do this kind of thing by hand, but in practice, there's no reason to, and while you're still learning calculus, the ability to check your answers is invaluable.
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Alright. Thank you, I shall check that out. I had been using Wolfram|Alpha for checking, but I had been having difficulty getting it to do partial derivatives and other topics that I've run into in Calc III.
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I thought I saw upon the stair a little man who wasn't there.
He wasn't there again today. Oh how I wish he'd go away.