If the area of the region bounded by the curves y^2-2y=-x and x+y=0 is A, then 8A is equal…

Question

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

Answer: 36

Accepted Answer

$x=2y-y^2$ and $x=-y$. Setting equal: $2y-y^2=-y \Rightarrow y(3-y)=0\Rightarrow y=0,3$.

$$A=\int_0^3\left[(2y-y^2)-(-y)\right]dy=\int_0^3(3y-y^2)\,dy=\left[\frac{3y^2}{2}-\frac{y^3}{3}\right]_0^3=\frac92.$$

$8A=36.$

Solutions · 1