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Old 7th November 2011, 06:22 AM   #1 (permalink)
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Multiple integration (volume)

i dun require the full working or any answer, just a simple question.
Qn:
By using an appropriate multiple integration method, compute the volume below the surface z= x^2y and above the regrion D which is in the first quadrant of the xy-plane and has the positive x-axis, positive y-axis, y = 16-x^2 and y = x^2 + 8 as its boundaries.

as shown in the image, tat would be region D which i got.
i intend to integrate dx followed by dy.

my fren and i got different answer because of what we placed in our "dx". may i know what is the upper and lower limit for dx?

also, just for checking, if i chose to integrate dy followed by dx is this correct?

thanks

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Old 7th November 2011, 08:00 PM   #2 (permalink)
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Re: Multiple integration (volume)

If you choose to integrate dy followed by dx, your expression is correct.
I have worked out both methods (dydx and dxdy) for your reference below. Hope it helps. Peace.


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Old 7th November 2011, 08:14 PM   #3 (permalink)
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Re: Multiple integration (volume)

thanks whitecorps....

btw, can i write this ?

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Old 7th November 2011, 08:22 PM   #4 (permalink)
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Re: Multiple integration (volume)

Originally Posted by KiLlErDeViL View Post
thanks whitecorps....

btw, can i write this ?
Definitely no. Your inner integral must have limits which are described in terms of y since you started off by using the operator dx (followed by dy), which means you must also start by scanning the boundaries in a horizontal fashion.


Last edited by whitecorp; 7th November 2011 at 08:23 PM.
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Old 7th November 2011, 08:24 PM   #5 (permalink)
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Re: Multiple integration (volume)

oic, back tracking abit, if i were to dydx, then i can do it within 1 double integration?

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Old 7th November 2011, 08:25 PM   #6 (permalink)
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Re: Multiple integration (volume)

hmm... i am sorry, but may i know how do i see, if i have to add 2 double integration ? or when can i just use 1 double integration?

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Old 7th November 2011, 08:32 PM   #7 (permalink)
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Re: Multiple integration (volume)

Your question deals with piecewise continuous functions (it involves two separate curves with totally different equations) so you have to segment your double integrals into two parts; if the boundary of the base is smooth and can simply be described by one single function with an upper bound y=g2(x) and lower bound y=g1(x) [or x=h2(y) and x=h1(y) ] , then you will only need one single double integral.

You can refer to this posting I wrote last year on my site:
http://www.whitegroupmaths.com/2010/...gration-2.html

The explanation for the single double integral thingy can be found in the first image at the top;
I have drawn up a full coloured diagram to make things clearer. Hope it helps. Peace.


Last edited by whitecorp; 7th November 2011 at 08:33 PM.
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Old 7th November 2011, 09:26 PM   #8 (permalink)
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Re: Multiple integration (volume)

thanks alot once again... was asking this qn coz i realized that my tutor did some of the tutorial question using only single double integral... just redid the tutorial question using 2x double integral and i got the same answer...

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Old 7th November 2011, 09:59 PM   #9 (permalink)
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Re: Multiple integration (volume)

No problem, am glad to be of assistance. Peace.

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