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Calculus/double integral, cardioid

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Expert: Alon Mandes - 10/20/2008


Question
Let R be the region in the xy-plane above the x-axis, bounded by the cardioid r=2+Cos(theta)

a. Find the area of R
b. SS(double integral signs) x/ x^2+y^2  dA

I used the uppercase S as an integral sign, the second integral has R at the bottom of it, any help would be greatly appreciated thank you!

Answer
Ok Heather, we have to integrate above the region R. But, since the
region is given in polar coordinates, we will use pollar integration
That means :
S=(R)∫∫dxdy=(R)∫∫ρdρdØ (" ρ(Ø)=2+cos(Ø) ")
Note that the area of a 2d region is defined as S=(R)∫∫dxdy.
Now, we need to set up limits of the integration :
1. Ø goes from 0 to 2π (" It's a closed loop ")
2. ρ goes from zero to it's normal equation 2+cos(Ø)
Let's ca;culate :
(R)∫∫ρdρdØ=∫dØ∫2+cos(Ø)dρ={0->2π}∫dØ[2+cos(Ø)]^2
I'll leave to you from here to proceed.

As for b.
(R)∫∫x/(x^2+y^2)dA. We use the parametrization:
1. x(ρ,Ø)=ρcos(Ø)
2. x^2+y^2=ρ^2
So, it will become :
=(R)∫∫[ρcos(Ø)]/ρ^2 {ρdρdØ}
=(R)∫∫cos(Ø)dρdØ
& the drill of this integration from here is similar to the former.
I'll leave it to you.

Alon.

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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M.A in Mathematics & Bs.c in Electronics.

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