Answer to Question #153362 in Differential Geometry | Topology for Tony

Question #153362

Shade the region of the xy-plane for which

a) (π‘₯2+𝑦2βˆ’16)(π‘₯2βˆ’4)≀0

b) (π‘¦βˆ’π‘₯)(𝑦2+π‘₯3)>0


1
Expert's answer
2021-02-24T07:33:35-0500

Before shading this region, let's study geometrically to which region it corres[onds. The first inequality is achieved, when two expressions ("x^2+y^2-16, x^2-4") have opposite signs. Now there is two cases : "x^2+y^2-16\\leq 0, x^2-4\\geq0", in this case the point should be in the circle centered at 0 of radius 4 (due to the first inequality) and it should have "|x|\\geq 2" due to the second case. The second case is when the point is outside the circle centered at 0 of radius 4, but it has "|x|\\leq 2". Thus we obtain a region like this :

We will study this region similarly to the previous one : this expression is positive if both the epressions "y-x, y^2+x^3" are of the same sign. There is two cases: "y>x, y^2>-x^3" or "y<x, y^2<-x^3". The first case corresponds to points that are higher than the straight line "y=x", and for the points "x<0" we have additional condition "|y|>\\sqrt{-x}^3". In the other case we have points that are lower than "y=x", we can't have points "x\\geq0" (as we can't have "y^2<0") and for "x<0" we have an additional condition "|y|< \\sqrt{-x}^3". Thus we obtain a region like this :

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