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    Area of Rectangle under a Parabola and Sectors in an Arc

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    Proving the area of a shaded rectangle under a parabola and then differentiating the expression.
    Minimizing the perimeter of sectors in an arc.

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    Solution Preview

    because parabola eqn. is given as:
    y = 3 - x^2
    Rectangle length is 2x and is symmetric about y axis, it means, the intersection points of parabola and rectangle are:
    (x, 3-x^2) and (-x, 3-x^2)
    therefore the area of shaded part:
    A = area of the rectangle = length * width
    width = ...

    Solution Summary

    The area of a rectangle under a parabola is calculated and the perimeter of a sector in an arc is minimized.