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  4. If the area of the region bounded by the curves y 2-2 y=-x, x+y=0 is A, then 8 A is equal to

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Q. If the area of the region bounded by the curves $y ^2-2 y=-x, x+y=0$ is $A$, then $8 A$ is equal to _________

JEE MainJEE Main 2023Application of Integrals

Solution:

$ y^2-2 y=-x $
$ \Rightarrow y^2-2 y+1=-x+1 $
$ (y-1)^2=-(x-1) $
$y=-x$
Points of intersection
$x^2+2 x=-x$
$ x^2+3 x=0 $
$ x=0,-3$
image
$ A=\int\limits_0^3\left(-y^2+2 y+y\right) d y$
$ =\frac{3 y^2}{2}-\left.\frac{y^3}{3}\right|_0 ^3=\frac{9}{2} $
$8 A=36$