Answer to Question #350717 in Calculus for Patchot

Question

Compute the area of the plane region bounded by the curve x=y²-2 and the line y=-x using integration along the y-axis.

Answer 1

Solution

Points of intersection of the given curves are solution of equation -y = y² – 2    => y² + y – 2 = 0   => Roots of this equation are y₁ = -2, y₂ = 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