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Math Help - Polar coordinates

  1. #1
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    Polar coordinates

    Sketch the curve given in polar coordinates by the equation

    r = 2 + cos Ѳ

    (a) Find the area of the region enclosed by the curve.
    (b) Giving answers to 2 decimal places, find the coordinates of the points on the curve where the tangent is parallel to the initial line.

    Iíve sketched the curve but what is the boundary? Then how can I solve these? Please help me.
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  2. #2
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    I've solved (a).
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  3. #3
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    Quote Originally Posted by geton View Post
    Sketch the curve given in polar coordinates by the equation

    r = 2 + cos Ѳ

    (a) Find the area of the region enclosed by the curve.
    (b) Giving answers to 2 decimal places, find the coordinates of the points on the curve where the tangent is parallel to the initial line.

    Iíve sketched the curve but what is the boundary? Then how can I solve these? Please help me.
    Not sure what you mean by "what is the boundary?" .... Do you mean the limits of integration?

    (a) Area enclosed by a polar curve r = r(\theta) is A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 \, d \theta.

    So the area of the region enclosed by your curve is r = r(\theta) is A = 2 \left(\frac{1}{2} \int_{0}^{\pi} (2 + \cos \theta)^2 \, d \theta \right) = \int_{0}^{\pi} (2 + \cos \theta)^2  \, d \theta.

    Not sure what you mean by "the tangent is parallel to the initial line". What is the initial line? And are the coordinates to be given as polar or Cartesian coordinates?

    Nevertheless, you'll probably need to get an expression for \frac{dy}{dx}. It will be a function of \theta. Do you know how to do this?

    Then solve \frac{dy}{dx} = m, where m is the gradient of the "initial line". This will give you the value(s) of \theta. Then sub the value(s) of \cos \theta into r = 2 + \cos \theta to get the corresponding value(s) of r => polar coordinates of required points on curve. These coordinates can obviously be converted into Cartesian coordinates quite easily.
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