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Volume

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Find the volume of the solid generated by revolving the region bounded by the given curves and line about the y-axis.

y=50-x^2
y=x^2
x=0

  • calculus -

    as I see it, the upper and lower boundries cross at 5,25, os the integral will be from x=0 t0 5

    so the dArea will be [(50-x^2)-x^2]dx
    and you will rotate that about the y axis, so the volume will be INT 2PI*xdA or

    int 2PI x(50-2x^2)dx

    so integrate that. I will be happy to check it.

  • Volume -

    Like all multiple integral problems, start with drawing a sketch of the bounding curves.
    I have done that for you for this time, see:
    http://img687.imageshack.us/img687/811/1342739949.png

    If curves intersect, find the intersection points. In this case, it is at (5,25).

    Then decide how you want to integrate, namely the order of integrating along x first, followed by y, or vice versa.

    Using the ring method, and set up the double integral:
    Volume
    =∫∫2πx dy dx
    y goes from x^2 to 50-x^2 and
    x goes from 0 to 5 (intersection point).

  • Volume -

    2Pi(500/3)

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