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January 30th, 2019, 01:02 PM   #1
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Cardioid and integration

How to find the integral of the function f(x,y) = y over the region D which is inside the cardioid r = 2 + 2 cosθ and outside the circle r=2?

I am unable to set the limits of the integrals. Please explain. The answer in my textbook for this comes out to be 22/3.

Please show me the answer with steps along with proper integral limits set and reason for choosing those limits?
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January 30th, 2019, 02:16 PM   #2
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Quote:
Originally Posted by shashank dwivedi View Post
How to find the integral of the function f(x,y) = y over the region D which is inside the cardioid r = 2 + 2 cosθ and outside the circle r=2?

I am unable to set the limits of the integrals. Please explain. The answer in my textbook for this comes out to be 22/3.

Please show me the answer with steps along with proper integral limits set and reason for choosing those limits?
let $R = 2(1+\cos{t})$, $r = 2$

note $R \ge r \implies \cos{t} \ge 0 \implies t \in \left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$

using symmetry ...

$\displaystyle A = 2\int_0^{\pi/2} \dfrac{R^2-r^2}{2} \, dt$

$\displaystyle A = \int_0^{\pi/2} 4(1+\cos{t})^2 - 4 \, dt$

btw, I don't agree w/ the book answer
Thanks from topsquark and shashank dwivedi
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January 31st, 2019, 12:25 AM   #3
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Quote:
Originally Posted by shashank dwivedi View Post
How to find . . .
Please give the exact wording of the question (using a picture if you can't manage the formatting). Your current wording is imprecise and skeeter misinterpreted it anyway.
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January 31st, 2019, 03:56 AM   #4
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I agree with your answer. Thank you for the clarification. The answer in the book is wrong, for if the answer was to be 22/3, then only upper only half plane should have been considered which is from 0 to pi/2.
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January 31st, 2019, 05:13 AM   #5
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What answer did you get for both parts (combined) of the region? Without knowing the precise original problem, it's impossible to know what answer is correct.
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January 31st, 2019, 05:27 AM   #6
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The question is from the Double Integration concept. I was asked to integrate by finding the limits as well for the following question.:

Find the Integral of f(x,y)=y over the region D which is inside the cardioid r = 2 + 2 cos theta and outside the circle r = 2.

I was confused with the limits to be chosen for the integration.

The answer given in my textbook is outer limit is chosen from 0 to pi/2 and inner limit is chosen from 2 to 2(1 + cos theta) and then this has been integrated by putting the jacobian and the answer comes out to be 22/3. As I understood, this is only for the upper half plane. For if the entire symmetric region was to be calculated, then the limit should have been from -pi/2 to pi/2 and the volume would have been zero. (As the negative and positive would have canceled out). However, since upper half is under consideration, I think 22/3 was to be calculated and the above question should have mentioned that the area under consideration is in the upper half plane.

Last edited by shashank dwivedi; January 31st, 2019 at 05:31 AM.
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January 31st, 2019, 06:46 AM   #7
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That was my thinking also, but I noticed that the question didn't explicitly state that it was a double integral and didn't even state what the integration was with respect to.
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January 31st, 2019, 01:12 PM   #8
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The question is perfectly clear and obviously cylindrical coordinates apply ($\displaystyle dA=rdrd\theta$). Cardioid is inside circle and symmetric with x axis.

$\displaystyle A=2\int_{0}^{\pi}\int_{2(1-\cos \theta)}^{2}rdrd\theta$

Edit: Whoops. Thought radius of circle was extent of cardioid. Use above but instead of starting $\displaystyle \theta$ at 0 start it where cardioid intersects circle, in this case $\displaystyle \pi$/2.

I was a little terse. Sorry.

Step 1) Draw a picture.
Step 2) Draw a picture.
Step 3) Draw a picture
Step 4) What is element of area? rdrd$\displaystyle \theta$.
Step 5) What is area? $\displaystyle \iint_{}^{}rdrd\theta.$
Step 6} What are limits of integration:
Draw a line at angle $\displaystyle \theta$ that intersects area you are looking for- in the area, where does it start and where does it end? Those are limits of integration for dr. Where does $\displaystyle \theta$ start and end? Those are limits of integration for d$\displaystyle \theta$.
Step 7) Do integration.

Last edited by skipjack; January 31st, 2019 at 02:42 PM.
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January 31st, 2019, 02:47 PM   #9
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You found one mistake, but you didn't spot that you'd misinterpreted the question, so you've still done the wrong thing.
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January 31st, 2019, 04:19 PM   #10
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Quote:
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You found one mistake, but you didn't spot that you'd misinterpreted the question, so you've still done the wrong thing.
I have a picture which makes everything perfectly clear. Unfortunately I can't figure out how to post it. If you select add image icon, it asks for url. What url?

The last time I tried to add an image there was an option below editing box for adding image and you could select it from your computer.
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