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Math Help - Finding the volume of a solid using it's equation?

  1. #1
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    Finding the volume of a solid using it's equation?

    A solid has, as its base, the circular region in the xy-plane bounded by the graph of x2 + y2 = 4. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is a quarter circle with one of its radii in the base.


    I don't know how to go about this? Do I make it a definite integral? I'm not sure. What am I supposed to get here?
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  2. #2
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    Re: Finding the volume of a solid using it's equation?

    one of its radii in the base
    I assume this means that the radius goes from one side of the circle to the other for the given value of x which the plane lies on.

    If the plane lies perpendicular to the x axis then is crosses the x axis at only one point.
    For a given point x where the plane crosses the x axis the "height" of the circle is \sqrt{4-x^2} (if you draw a 2D diagram of the circle with the y axis in the upwards direction then it is more clear what I mean by height)
    The radius of the quarter circle at this point will be 2\sqrt{4-x^2}.
    The area of the quarter circle will be \frac{1}{4} \pi r^2= \frac{1}{4} \pi (2\sqrt{4-x^2})^2.

    The volume of a small slice of the quarter circle is dV. dV=\frac{1}{4} \pi (2\sqrt{4-x^2})^2 dx.

    Integrate the volume between the values of x where a plane crosses the circle to get the volume of the solid
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