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Trouble finding curve intersection

  1. Mar 12, 2008 #1
    [SOLVED] trouble finding curve intersection

    1. The problem statement, all variables and given/known data
    Sketch the area of the region bounded by the curves [tex]y^2+4x=0[/tex] and [tex]y=2x+4[/tex]. Set up two integrals, one with respect to x and one with respect to y, for finding the area of the region. Evaluate one of the integrals to find the area.

    2. Relevant equations

    3. The attempt at a solution
    First I rearrange the equations of the curves to isolate both x & y for each.
    [tex]y^2+4x=0 \Longrightarrow y^2=-4x \Longrightarrow \sqrt{y^2}=\sqrt{-4x} \Longrightarrow y=2\sqrt{-x},-2\sqrt{-x}[/tex]
    [tex]y^2+4x=0 \Longrightarrow x=-\frac{1}{4}y^2[/tex]

    [tex]y=2x+4 \Longrightarrow x=\frac{y-4}{2} \Longrightarrow x=\frac{1}{2}y-2[/tex]

    Next I must find the domain of the integrals by setting y=y.
    [tex]-\frac{1}{4}y^2=\frac{1}{2}y-2 \Longrightarrow \frac{1}{4}y^2+\frac{1}{2}y-2 \Longrightarrow \frac{1}{2}(y-2)(y+4) \Longrightarrow y=2,-4[/tex]

    This is where I'm running into trouble, when I set x=x.
    [tex]2\sqrt{-x}=2x+4[/tex] [tex]-2\sqrt{-x}=2x+4[/tex]
    I know the points of intersection are -1 & -4 respectively but I can't seem to tease those out of the equations.

    [tex]2\sqrt{-x}=2x+4 \Longrightarrow 2x^\frac{1}{2}+2x+4 \Longrightarrow 2(x^\frac{1}{2}+x+2)[/tex]
    [tex]-2\sqrt{-x}=2x+4 \Longrightarrow -2x^\frac{1}{2}+2x+4 \Longrightarrow 2(-x^\frac{1}{2}+x+2)[/tex]
    I'm not sure what to do from here. I can't factor this and if I square it and apply the quadratic equation I get a negative under the sqrt. Some advice would be greatly appreciated. Cheers.
    Last edited: Mar 12, 2008
  2. jcsd
  3. Mar 12, 2008 #2
    you have
    y^2 + 4x = 0
    y = 2x + 4

    now, y^2 + 4x = 0 => y^2 = -4x => -2x = (y^2)/2

    also, y = 2x + 4 => -2x = 4 - y.

    So 4 - y = y^2/2, so y^2 + 2y - 8 = 0, so y = - 4 or y = 2, plugging these in
    gives that your points of intersection are (-4, -4), (-1, 2).

    now just draw the picture and write down your integrals.
  4. Mar 12, 2008 #3
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