A calculus problem by Hobart Pao

Calculus Level 4

Given the parabolas \( y = x^2, y=-x^2, x=y^2, x=-y^2 \), an osculating circle can be drawn on each parabola at the origin.

You now have 4 osculating circles, and they create four areas of intersection.

The total area of these 4 intersections is equal to the area of intersection of the two circles \( r= a \cos \theta \) and \( r=a \sin \theta \). Find \(a \).

Clarification:

The osculating circle to \( \vec{r}(t) \) at time \( t \) is the circle tangent to \( \vec{r}(t) \) at time \( t \) with radius \( \dfrac{1}{\kappa (t)} \), where \( \kappa(t) \) is the curvature of \( \vec{r}(t) \) at time \( t \).

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