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Math Help - question on the volume of a pool in the shape of an elipse

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    question on the volume of a pool in the shape of an elipse

    As viewed from above, a swimming pool has the shape of the ellipse
    x^2/25 +y^2/16=1
    The cross sections perpendicular to the ground and parallel to the y-axis are squares.
    Find the total volume of the pool. (Assume the units of length and area are meters
    and square meters respectively.)
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    Re: question on the volume of a pool in the shape of an elipse

    Quote Originally Posted by Michellemo View Post
    As viewed from above, a swimming pool has the shape of the ellipse
    x^2/25 +y^2/16=1
    The cross sections perpendicular to the ground and parallel to the y-axis are squares.
    Find the total volume of the pool. (Assume the units of length and area are meters
    and square meters respectively.)
    side of a square = 2x

    area of the square = 4x^2

    x^2 = 25\left(1 - \frac{y^2}{16}\right)

    V = 2\int_0^4 A(y) \, dy

    finish it
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    Re: question on the volume of a pool in the shape of an elipse

    So for a given y, the two x values are x= \pm 5\sqrt{1- y^2/16}= \pm\frac{5}{4}\sqrt{16- y^2} and the length of that segment is \frac{5}{2}\sqrt{16- y^2}. Since "cross sections perpendicular to the ground and parallel to the y-axis are squares", the depth at that y is also \frac{5}{2}\sqrt{16- y^2} and so the cross section area is the square of that, \frac{25}{4}(16- y^2). Taking the thickness of an "infinitesmal slab" to be [itex]dy[/tex], the volume of such a slab is [itex]\frac{25}{4}(16- y^2)dy[/itex] and the total volume is the integral of that.
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