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About Alon Mandes
Expertise
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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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.
Education/Credentials
M.A in Mathematics & Bs.c in Electronics.
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You are here: Experts > Teens > Homework/Study Tips > Calculus > Integals
Expert: Alon Mandes - 11/3/2009
Question
Hi Alon,
the question asks to calculate the integral
Z=(integral)R (x^2-y^2)dR
Where R is the region of a circle of radius 1 and radius 2, axis x and the line y=x in the 1st quadrant.
Thanks
Answer
The region R can be described in polar form as : 1≤ρ≤2 %26 0≤θ≤π/4 . So we sill perform the
transformation : x=ρcos(θ) %26 y=ρsin(θ) . The Jacobean of this trans is |J|=ρ . Therefore
dR will become ρdρdθ . Thus,
∫∫(x²-y²) dR =
R
π/4 2
∫ ∫ [ρ²cos²(θ)-ρ²sin²(θ)]ρdρdθ =
0 1
π/4 2
∫ ∫ ρ³[cos²(θ)-sin²(θ)] dρdθ .
0 1
According to trigonometric identities : cos²(θ)-sin²(θ)=cos(2θ) . Therefore :
π/4 2
∫ ∫ ρ³[cos²(θ)-sin²(θ)] dρdθ =
0 1
π/4 2
∫ ∫ ρ³cos(2θ) dρdθ =
0 1
π/4 2
∫ cos(2θ)dθ * ∫ ρ³dρ =
0 1
(1/2)sin(2θ) { from 0 to π/4 } * (1/4)ρ^4 { from 1 to 2 } =
(1/2)sin(π/2) * [(1/4)2^4 - (1/4)] = (1/2)*(1/4)[16-1] = 15/8 .
Alon.
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