Cardioid and integration

Apr 2017
87
1
India
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?
 

skeeter

Math Team
Jul 2011
3,361
1,851
Texas
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
 
  • Like
Reactions: 2 people

skipjack

Forum Staff
Dec 2006
21,481
2,470
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.
 
Apr 2017
87
1
India
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.
 

skipjack

Forum Staff
Dec 2006
21,481
2,470
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.
 
Apr 2017
87
1
India
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. :confused:

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:

skipjack

Forum Staff
Dec 2006
21,481
2,470
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.
 
Mar 2015
1,720
126
New Jersey
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 a moderator:

skipjack

Forum Staff
Dec 2006
21,481
2,470
You found one mistake, but you didn't spot that you'd misinterpreted the question, so you've still done the wrong thing.
 
Mar 2015
1,720
126
New Jersey
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.