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Math Help - Cartesian equations, areas of regions, and infinite sequences and series help needed.

  1. #1
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    Exclamation Cartesian equations, areas of regions, and infinite sequences and series help needed.

    Please help me with whatever you're able to. And try to explain what's going on in the solving process. I want to understand this, not just get answers. Thank you so so much!!!

    identify the curve by finding a cartesian equation for the curve:

    r cos (theda) = 1

    Find a polar equation for the curve represented by the given Cartesian equation:

    x + y = 9

    Find the area of the region that is bounded by the given curve and lies in the specified sector
    r = e^(theta/2), pi less than or equal to theta less than or equal to 2pi

    find the area enclosed by the curve
    r = 2-sin(theta)

    Find the area of the region that lies inside both curves
    r = 1 + cos(theta), r = 1-cos(theta)

    Determine whether the sequence converges or diverges. If it converges, find the limit.
    a(n)= [(-1)^n * n^3]/[n^3 + 2n^2 +1]

    {arctan 2n]

    [ln(n)]/[ln(2n)]

    Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
    a(n)=ne^-n
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  2. #2
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    To start with, you should know that x = r\cos{\theta}, y = r\sin{\theta} and x^2 + y^2 = r^2.
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  3. #3
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    so then,

    so it would be x=rcos(th) and y=rcos(th), giving the line x=1 for the first question?

    and for the second, x+y=9, would x^2 + y^2 = r^2 would give, r=sqrt(x^2 + y^2) and theta=arctan (y/x)? and that would be, what? rcos(th) + rsin(th)=9?

    And how do I approach the areas and series problems?
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  4. #4
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    Yes, x = 1 is correct for the first equation.

    For the second, substitute x = r\cos{\theta} and y = r\sin{\theta}. The best you can do is to take a common factor of r.
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