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CavalieriEllipse - x 2 a 2 y 2 b 2 = 1 Compare this area...

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Cavalieri’s Determination of the Area of an Ellipse The principle is the following. Suppose that two planar figures have the same height and at the same level the cross-sectional lengths are in the same ratio r . Then the areas are in the same ratio r . Cavalieri applied this to determine the area of an ellipse. In standard form, the equation of an ellipse is
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Unformatted text preview: x 2 a 2 + y 2 b 2 = 1 . Compare this area with that of a circle with radius a whose area is ! a 2 . x 2 + y 2 = a 2 At x the cross-sectional lengths are 2 y = b a a 2 ! a 2 and 2 y = a 2 ! x 2 , respectively. Thus the ratios of these lengths is b a . From this we get that the area of the ellipse is A ellipse = b a ! " a 2 = ab . a b a...
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