Find the area bounded by $x=-y^2$ and $y=x+2$.

Question

Question

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Find the area bounded by $x=-y^2$ and $y=x+2$.

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My Attempt

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I know it is a very simple question to ask on MSE, but I don't know why I get stuck. If you trace the graph, then the point of intersection will be $(-4,2)$ and $(-1,1)$.The problem is that the parabola is not a function, hence it has two corresponding $y$ for one $x$ and I think that's the reason why I get stuck. I tried it assuming that area between $y^2=x$ and $y=-x+2$, but again got stucked.

I dont know what's troubling me.

Please help.

Answer 1

HINT

In that case is always a good idea to make a sketch

which suggests an easier calculation by the following set up

$$A=\int_{-2}^1 f(y) dy$$

Answer 2

$$\int_{-2}^{1}\int_{y-2}^{-y^2} dxdy$$