Circumparabolic Regions Inside a Unit Circle

circumparabolic regions

A circumparabolic region is found between a circle and a parabola, with the circle being chosen to include the vertex and x-intercepts of the parabola used, with the circle, to define the two circumparabolic regions for a given parabola-circle pair. There are four such regions shown above, rather than only two, because two parabolas are used above. The formulae for the parabolas, as well as the circle, are shown.

A puzzle which I will not be solving, I suspect, until I learn more integral calculus: what fraction of the circle’s area is shown in yellow?

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RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things, a large portion of which are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.

10 thoughts on “Circumparabolic Regions Inside a Unit Circle”

    1. Well, formally speaking, you can find the area between a parabola and the x-axis by exhaustion, so that’s 4/3ds times the area of the triangle bounded by the points (-1, 0), (0, 1), and (1, 0). And you know the area of the circle. But that’s just using integration by a softer name.

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      1. I would strongly recommend it since taking an anti-derivative of a simple polynomial was
        2nd semester stuff ( back in 1974 anyway!)

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