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Math Help - Centroid & Volume of region

  1. #1
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    Centroid & Volume of region

    The region in the first quadrant bounded by the graphs of  y=x and y=\frac{x^2}{2} is rotated around the line  y=x Find the centroid of the region and the volume of the solid of revolution.
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  2. #2
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    Rotate the coordinate axes \pi/4.
    The transformation will make a new graph.
    Now simplify find the volume of the region around the x-axis.
    (I will do this problem latter. Right now I am using one of my many computers. This one does not have a gprahing program.)
    ---
    You have,
    2y=x^2 this describes your curve.
    From conics you should be familar that a rotation of \pi/4 transforms,
    (x,y)\to (\cos \pi/4\cdot x-\sin \pi/4\cdot y,\sin \pi/4 \cdot x+\cos \pi/4 \cdot y)
    Equivalently,
    (x,y)\to \left( \frac{\sqrt{2}}{2} x-\frac{\sqrt{2}}{2} y, \frac{\sqrt{2}}{2} x+\frac{\sqrt{2}}{2}y\right)
    Thus,
    2\left( \frac{\sqrt{2}}{2} x+\frac{\sqrt{2}}{2}y\right)=\left( \frac{\sqrt{2}}{2} x-\frac{\sqrt{2}}{2} y\right)^2
    Thus,
    \sqrt{2} x+\sqrt{2}y=\frac{1}{2}x^2-\frac{1}{2} xy+\frac{1}{2}y^2
    2\sqrt{2} x+2\sqrt{2} y=x^2-xy+y^2
    Now the problem is that this curve is not a function (if I only had a graph) and the shell formula needs to curve to be a function. So I am going to divide this curve into two curves that are functions and add their rotational volume about the y-axis. In order to do that I will first need to bring this to the form y=f(x). To do that I need to solve for y.
    First rewrite as,
    y^2-y(x+2\sqrt{2})+(x^2-2\sqrt{2} x)=0
    Thus,
    y=\frac{-x-2\sqrt{2}\pm \sqrt{x^2+4x\sqrt{2}+8-4x^2+8x\sqrt{2}}}{2}
    (TO BE CONTINUED)
    Last edited by ThePerfectHacker; August 31st 2006 at 05:01 PM.
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  3. #3
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    I hope everyone had a nice holiday weekend, at least those stateside anyway. Just wanted to bump this to the top in hopes of the continuation....
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  4. #4
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    Please, do not make me.
    It would be a mess.
    ---
    Why do you even need it?
    ---
    Attached Thumbnails Attached Thumbnails Centroid & Volume of region-picture2.gif  
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  5. #5
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    Well, since you asked. I am taking a cracker-jack on-line Calc I class. This is the last question on an open-book test. I have answered the others, some with the help of this forum. I am averaging a whopping 72% in the class, and passing is all I care about right now Anyway, I can't really afford to leave this one blank, the adjunct usually gives partial credit for at least getting part of it right!
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  6. #6
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    Quote Originally Posted by Zeppelin
    Well, since you asked. I am taking a cracker-jack on-line Calc I class. This is the last question on an open-book test. I have answered the others, some with the help of this forum. I am averaging a whopping 72% in the class, and passing is all I care about right now Anyway, I can't really afford to leave this one blank, the adjunct usually gives partial credit for at least getting part of it right!
    Who to, ImPerfectHacker?

    RonL
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