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Math Help - Volumes by Cross-Sections

  1. #1
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    Volumes by Cross-Sections

    A solid has at its base the region bounded by the parabola x-y^2 = -8 and the left branch of the hyperbola x^2-y^2-4=0. The vertical slices perpendicular to the x-axis are squares. Find the volume of the solid. This is the full problem.

    I was able to find out the points of intersection, which were x=-3 and x=4. I am not sure what the cross-sectional area is.
    Last edited by dalbir4444; January 31st 2010 at 01:41 PM.
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by dalbir4444 View Post
    A solid has at its base the region bounded by the parabola x-y^2 = -8and the left branch of the hyperbola x^2-y^2-4=0. The vertical slices perpendicular to the x-axis are squares. Find the volume of the solid.
    Could you please post the problem as it is written and show your attempt at the solution?
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  3. #3
    MHF Contributor
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    hi dalbir4444,

    the left side of the hyperbola is x=-\sqrt{y^2+4}

    You've identified the x co-ordinate of the points of intersection.
    x=4 is unnecessary, as the parabola intersects the rightside part there.

    To calculate the area of the bounded region, you only need the area of the part above the x-axis and multiply by 2.

    If you also calculate where the parabola and hyperbola cut the x-axis to the left of the y-axis, then you can integrate under the parabola from x=-8 to -3
    and integrate under the hyperbola from x=-3 to x=-2.

    That will be one way to calculate the area between the curves.

    I can't visualise the object itself however, as your question mentions a square.
    Attached Thumbnails Attached Thumbnails Volumes by Cross-Sections-parabola-hyperbola.jpg  
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