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How to solve a surface double integral?

  • Thread starter al_ex
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Problem Statement
surface double integral, sphere, cardioid
Relevant Equations
how do I plan the problem?
Hi I´d like a suggestion about a surface double integral. If I have a sphere x^2+y^2+z^2=4 is on the top of a cardioid r=1-cosθ. The problem is when I solve the integral I got a inverse sine when the answer is a natural logarithm (ln)
 

LCKurtz

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Problem Statement: surface double integral, sphere, cardioid
Relevant Equations: how do I plan the problem?

Hi I´d like a suggestion about a surface double integral. If I have a sphere x^2+y^2+z^2=4 is on the top of a cardioid r=1-cosθ. The problem is when I solve the integral I got a inverse sine when the answer is a natural logarithm (ln)
Hi al_ex. Welcome to PF. You have to show some effort before anyone will help you. I would suggest you take a look at this insight article:
Then try parameterizing your surface in cylindrical coordinates and come back if you get stuck. Also you might look at the LaTeX guide for typing equations:
 
I am parameterizing dz/dx and dz/dy I got this r/(4-r^2)^(1/2) dr dθ and my parameters are from 0 to 1-cosθ and from 0 to π but when I integrate r/(4-r^2)^(1/2) dr, I got (4-r^2)^(1/2) and when I put the values 0 and 1-cosθ is where I´m starting to get a troubles because I can´´ not get in the second integration the natural logarithm I think Im planning wrongly my equations. I really need help. The answer of this problem is 8 (π -(2)^1/2-ln<(2)^1/2+1>) I dont think that the book´'s answer is wronged.
 

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LCKurtz

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When I parameterize it with ##r## and ##\theta## I get$$
A=\int_0^{2\pi}\int_0^{1-\cos\theta}~\frac{2r}{\sqrt{4-r^2}}~drd\theta$$
for which Maple gives ##8\pi -8\sqrt 2 +4 \ln(3-2\sqrt 2)##. I think your ##\theta## limits are wrong and anything else is difficult to guess without seeing your steps.
 

LCKurtz

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Let me add...in a more direct response to your integration problem, while Maple easily evaluates ##\int_0^{2\pi}\sqrt{4-(1-\cos\theta)^2}~d\theta##, it gives a horribly messy antiderivative. It's no wonder you are having trouble integrating it. Once you have it properly set up I wouldn't waste any more time trying to evaluate it unless you have an appropriate software program to do it for you.
 
what do you suggest me? what can I do? this problem has any solution? can you give me a hint? please...
 

LCKurtz

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In post #4 I have given you a correct setup and a correct answer to check your work against. Assuming you agree with that, you have the problem set up correctly. Like I said in post #5, actually working out the antiderivative looks to be complicated and time consuming and a waste of time. I don't plan to waste any more time trying to do it. Ask your teacher how he/she would proceed.
 

vela

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While Maple easily evaluates ##\int_0^{2\pi}\sqrt{4-(1-\cos\theta)^2}~d\theta##, it gives a horribly messy antiderivative.
It doesn't look too bad to do by hand starting with the half-angle trig identity. I got it down to an integral of the form
$$\int \sqrt{1+u^2}\,du.$$ You can look this last one up in a table of integrals.
 

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