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Math Help - Double integral in polar coordinates

  1. #1
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    Double integral in polar coordinates

    Evaluate the integral by changing to polar coordinates. ∫∫[D] x dA. Where D is the region in the first quadrant that lies bewteen the circles x^2+y^2=4 and x^2+y^2=2x.


    Please, explain how can you find the limit for "r" ..
    first circle has center (0,0), radius = 2
    second circle has center (1,0), radius = 1
    I know that
    but where is D !!
    "BETWEEN TWO CIRCLES" << Omg, i cant figure it out =(

    And this is not for my homework
    Actually, we dont hae homeworks in our university =D
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  2. #2
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    Quote Originally Posted by TWiX View Post
    Evaluate the integral by changing to polar coordinates. ∫∫[D] x dA. Where D is the region in the first quadrant that lies bewteen the circles x^2+y^2=4 and x^2+y^2=2x.


    Please, explain how can you find the limit for "r" ..
    first circle has center (0,0), radius = 2
    second circle has center (1,0), radius = 1
    I know that
    but where is D !!
    "BETWEEN TWO CIRCLES" << Omg, i cant figure it out =(

    And this is not for my homework
    Actually, we dont hae homeworks in our university =D
    Cartesian equation: x^2+y^2=4. Polar equation: r = 2.

    Cartesian equation: x^2+y^2=2x. Polar equation: r^2 = 2 r \cos \theta \Rightarrow r = 2 \cos \theta.

    Note: x^2+y^2=2x \Rightarrow x^2 - 2x + y^2 = 0 \Rightarrow (x - 1)^2 - 1 + y^2 = 0 \Rightarrow (x - 1)^2 + y^2 = 1.

    I suggest you draw the two circles (that ought to be a trivial thing to do) and shade the required region. Then set up the polar integration.
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    I know theta between 0 and pi/2
    I know how to draw the two circles
    and i know the circle with radius 1 will lies inside the other circle
    BUT the problem is I cant figure this >> "LIES BETWEEN THE TWO CIRCLES"
    where is the region which lies between two circle!!
    can you show it to me?
    and another question please:
    r from 2cosθ to 2
    but why it not from 2 to 2cosθ ??!
    sorry, but i have little problems in this section =(
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    Lies between the two circles = inside both circles
    ?!
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    So the region will be the half of the small circle above the x-axis
    ok
    then theta lies between 0 and pi/2
    Still good !
    But the problem is the limits of r!
    this region is bounded by two polar curves
    r = 2
    and r = 2cosθ
    Fantastic! i got it.. but still i have a small problem
    why is it from 2cosθ to 2
    isnt it from 2 to 2cosθ ??
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  6. #6
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    Quote Originally Posted by TWiX View Post
    So the region will be the half of the small circle above the x-axis
    ok
    then theta lies between 0 and pi/2
    Still good !
    But the problem is the limits of r!
    this region is bounded by two polar curves
    r = 2
    and r = 2cosθ
    Fantastic! i got it.. but still i have a small problem
    why is it from 2cosθ to 2
    isnt it from 2 to 2cosθ ??
    Draw the diagram like I said to do. It should be crystal clear that you are integrating from the inner circle to the outer circle and I have given you the polar equation of those two curves. I suggest you go back and review the formula for the area between two polar curves.
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    Quote Originally Posted by TWiX View Post
    Lies between the two circles = inside both circles
    ?!
    I doubt that. What you have is a circle with center at the origin and radius 2 and a circle with origin at (0, 1) and radius 1. The second circle lies completely inside the first so the area "inside both circles" would be just the area of the smaller circle, \pi. "Lies between the two circles" means just that- inside the larger circle but outside the smaller.
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  8. #8
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    graph ...
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    Quote Originally Posted by mr fantastic View Post
    Draw the diagram like I said to do. It should be crystal clear that you are integrating from the inner circle to the outer circle and I have given you the polar equation of those two curves. I suggest you go back and review the formula for the area between two polar curves.
    lol am drawing in the xy-plane ;s
    Ok thanks

    But what is the relation-ship between the area in polar coordinates and this integral!!!
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  10. #10
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    Quote Originally Posted by HallsofIvy View Post
    I doubt that. What you have is a circle with center at the origin and radius 2 and a circle with origin at (0, 1) and radius 1. The second circle lies completely inside the first so the area "inside both circles" would be just the area of the smaller circle, \pi. "Lies between the two circles" means just that- inside the larger circle but outside the smaller.
    its a stupid sentence
    i hate it

    thanks
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    Quote Originally Posted by skeeter View Post
    graph ...
    so the region is the upper one?
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  12. #12
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    Quote Originally Posted by TWiX View Post
    so the region is the upper one?
    yes ... between the red and blue in quad I
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