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Math Help - Volume of Revolution question!

  1. #1
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    Lightbulb Volume of Revolution question!

    The normal to the curve y=3x^2+2 at the point B(1,5) meets the x-axis at the point C.
    The finite region bounded by the curve, the line BC and the y-axis is rotated through 2pi radians about the x-axis
    Find the volume of the solid of revolution generated?


    So i really can't get my head around this!
    But this is what i did..


    V = ∫[0 to 1] πy^2 dx
    = ∫[0 to 1] π(3x^2 + 2)^2 dx
    = ∫[0 to 1] π(9x^4 + 12x^2 + 4) dx
    = π[(9/5)x^5 + (12/3)x^3 + 4x)]|[0 to 1]
    = π[(9/5)x^5 + 4x^3 + 4x)]|[0 to 1]
    = π[(9/5)*1 + 4*1 + 4*1]
    = π[(9+40)/5]
    = 49π/5

    i do believe that theres more to it and that's not simply the answer, but i really don't have clues!
    please help!
    thanks
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Volume of Revolution question!

    For x = 0 to x = 1, the disks do have a radius described by the given parabola, but you need to let the disks have a radius given by the normal curve from x = 1 until the x=intercept of the normal line. Can you give the equation of the normal line?
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  3. #3
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    Re: Volume of Revolution question!

    Quote Originally Posted by MarkFL View Post
    For x = 0 to x = 1, the disks do have a radius described by the given parabola, but you need to let the disks have a radius given by the normal curve from x = 1 until the x=intercept of the normal line. Can you give the equation of the normal line?
    So is the normal equation
    y=-x+1 ?

    I still don't quite grasp it! :/
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: Volume of Revolution question!

    Quote Originally Posted by Luke2806 View Post
    So is the normal equation
    y=-x+1 ?

    I still don't quite grasp it! :/
    No, you want to find the slope of the tangent line at the given point, the use the negative reciprocal of that, and then use the point-slope formula to determine the normal line. What is the slope of the tangent line?
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