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The area bounded by the circle $x^2 +y^2=8$ , the parabola $x^2 =2y$, and the line $y=x$ is..?

The integration is pretty easy to do but I'm having a difficult time finding the area between three curves in the above figure. What would you define in the following figure as the area between the three curve? / what is the criteria for it?

tryst with freedom
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  • You are right to be confused because the question is ambiguous. It could be any of those areas without more restrictions. – Ninad Munshi Feb 01 '21 at 19:02
  • @NinadMunshi What else conditions should be provided to make the question not ambigous? – tryst with freedom Feb 01 '21 at 19:05
  • If the bounding curves were not in form of equations but in form of inequalities, it would be clear. In that case, each inequality would define a subset of a plane and the area would be their intersection. – Hume2 Feb 01 '21 at 19:07
  • Anything that distinguishes the areas like quadrants, inequalities in the original equations, etc – Ninad Munshi Feb 01 '21 at 19:07
  • @NinadMunshi does $y \geq 0 $ suffice? – tryst with freedom Feb 01 '21 at 19:18
  • It would not because there are still $5$ different regions it could be referring to ($4$ inside the circle and $1$ outside off to the upper right), assuming of course all of the bounding curves "touch" the region in question. – Ninad Munshi Feb 01 '21 at 19:20
  • I think we have to find area inside the circle between parabola(inside the U and circle) then subtract area between line and parabola(inside the U) from it. – sirous Feb 01 '21 at 19:30

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For each of the potential boundaries you have to specify which side of the boundary contains the area you care about. For the circle it's natural to think about the inside. For the parabola it's natural to think about the convex region ("inside the U").

There is no natural way to distinguish one side of the line from the other, so you have to specify that explicitly.

You could also say "beneath the parabola" or "outside the circle" if that was what mattered to you.

Algebraically, you would specify one side of a boundary with an inequality.

Ethan Bolker
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