Question #350717

Compute the area of the plane region bounded by the curve x=y^{2}-2 and the line y=-x using integration along the y-axis.

Expert's answer

**Solution**

Points of intersection of the given curves are solution of equation -y = y^{2} – 2 => y^{2 }+ y – 2 = 0 => Roots of this equation are y_{1} = -2, y_{2} = 1

So area to be find is the area bounded by curves x = y² – 2 and x = -y (-y> y² – 2 for -2<y<1)

"A=\\int_{-2}^{1}\\left(-y-y^2+2\\right)dy=\\left(2y-\\frac{1}{2}y^2-\\frac{1}{3}y^3\\right)\\left|\\begin{matrix}1\\\\-2\\\\\\end{matrix}\\right."

A = (2 – 1/2 – 1/3) + (4 + 2 – 8/3) = 8 – 1/2 – 9/3 = 4.5

**Answer**

A = 4.5

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