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Math Help - Polar to Cartesian.

  1. #1
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    Polar to Cartesian.

    In my calculus class, I chose to doa project on finding the area inside a polar curve and then revolving the curve around the x-axis in order to find the surface area and the volume of the solid. Unfortunately, I got stuck on converting r=2.5sin(3θ) into Cartesian form. Is this possible and if so how is it done?
    Last edited by mr fantastic; May 31st 2010 at 01:25 PM. Reason: Edited post title (of course yuo want help!)
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    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by phantomfn8 View Post
    In my calculus class, I chose to doa project on finding the area inside a polar curve and then revolving the curve around the x-axis in order to find the surface area and the volume of the solid. Unfortunately, I got stuck on converting r=2.5sin(3θ) into Cartesian form. Is this possible and if so how is it done?

    I thought it was just another circle.
    I believe it's a 3 leaf petal now.
    Last edited by matheagle; May 30th 2010 at 05:35 PM.
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    So is it possible to convert to cartesian form?
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  4. #4
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    You can either convert it because you know the cartesian equation of the circle matheagle described, or you can use the formula below
    x=rcos(\theta)
    y=rsin(\theta)

    These can be used to convert any polar coordinates (r,\theta) to cartesian coordinates x,y, but it looks messy....id work out the equation of the circle instead.
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  5. #5
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    Quote Originally Posted by phantomfn8 View Post
    In my calculus class, I chose to doa project on finding the area inside a polar curve and then revolving the curve around the x-axis in order to find the surface area and the volume of the solid. Unfortunately, I got stuck on converting r=2.5sin(3θ) into Cartesian form. Is this possible and if so how is it done?
    a single "petal", or is the whole thing to be rotated about the x-axis?

    for the volume, you might consider Pappus's Centroid Theorem -- from Wolfram MathWorld
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  6. #6
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    sin(\theta+ \phi)= sin(\theta)cos(\phi)+ cos(\theta)sin(\phi)\
    and
    cos(\theta+ \phi)= cos(\theta)cos(\phi)- sin(\theta)sin(\phi).

    In particular,
    sin(2\theta)= sin(\theta+ \theta) = sin(\theta)cos(\theta)+ cos(\theta)sin(\theta)= 2sin(\theta)cos(\theta)
    and
    cos(2\theta)= cos(\theta+ \theta) = cos(\theta)cos(\theta)- sin(\theta)sin(\theta)= cos^2(\theta)- sin^2(\theta)

    Now,
    sin(3\theta)= sin(2\theta+ \theta)= sin(2\theta)cos(\theta)+ cos(2\theta)sin(\theta) = 2sin(\theta)cos^2(\theta)+ (cos^2(\theta)- sin^2(\theta))sin(\theta) = 2sin(\theta)cos^2(\theta)+ cos^2(\theta)sin(\theta)- sin^3(\theta) = 3sin(\theta)cos^2(\theta)- sin^3(\theta).

    So r= .5 sin(3\theta)= .5(3 sin(\theta)cos^2(\theta)- sin^3(\theta)

    Multiply both sides by r^3 to get
    r^4= 1.5 (r sin(\theta)(r^2 cos^2(\theta))- .5 (r^3 sin^2(\theta))

    r^2= x^2+ y^2 so r^4= (x^2+ y^2)^2 while r cos(\theta)= x and r sin(\theta)= y.

    (x^2+ y^2)^2= 1.5 xy^2- .5 y^3
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    I intended odor the entire curve to be rotated around the x-axis. How do you find the volume of parametric equations rotated around the axis?
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  8. #8
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    Quote Originally Posted by HallsofIvy View Post
    sin(\theta+ \phi)= sin(\theta)cos(\phi)+ cos(\theta)sin(\phi)\
    and
    cos(\theta+ \phi)= cos(\theta)cos(\phi)- sin(\theta)sin(\phi).

    In particular,
    sin(2\theta)= sin(\theta+ \theta) = sin(\theta)cos(\theta)+ cos(\theta)sin(\theta)= 2sin(\theta)cos(\theta)
    and
    cos(2\theta)= cos(\theta+ \theta) = cos(\theta)cos(\theta)- sin(\theta)sin(\theta)= cos^2(\theta)- sin^2(\theta)

    Now,
    sin(3\theta)= sin(2\theta+ \theta)= sin(2\theta)cos(\theta)+ cos(2\theta)sin(\theta) = 2sin(\theta)cos^2(\theta)+ (cos^2(\theta)- sin^2(\theta))sin(\theta) = 2sin(\theta)cos^2(\theta)+ cos^2(\theta)sin(\theta)- sin^3(\theta) = 3sin(\theta)cos^2(\theta)- sin^3(\theta).

    So r= .5 sin(3\theta)= .5(3 sin(\theta)cos^2(\theta)- sin^3(\theta)

    Multiply both sides by r^3 to get
    r^4= 1.5 (r sin(\theta)(r^2 cos^2(\theta))- .5 (r^3 sin^2(\theta))

    r^2= x^2+ y^2 so r^4= (x^2+ y^2)^2 while r cos(\theta)= x and r sin(\theta)= y.

    (x^2+ y^2)^2= 1.5 xy^2- .5 y^3
    Thank you soo much! Now would ou know how to find the volume of the graph when rotate around the x-axis?
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