JEE MAIN - Mathematics (2023 - 24th January Evening Shift - No. 21)
If the area of the region bounded by the curves $$y^2-2y=-x,x+y=0$$ is A, then 8 A is equal to __________
Answer
36
Explanation
Area enclosed by
$$ \begin{aligned} & y^{2}-2 y=-x \\\\ & x+y=0 \end{aligned} $$
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Area $=\int_{0}^{3}\left(2 y-y^{2}\right)-(-y) d y$
$$ =\int_{0}^{3}\left(3 y-y^{2}\right) d y $$
$$ \begin{aligned} & \left.=\frac{3 y^{2}}{2}-\frac{y^{3}}{3}\right]_{0}^{3} \\\\ & =\frac{27}{2}-9 \\\\ & =\frac{27-18}{2}=\frac{9}{2}=A \end{aligned} $$
$8 A=\frac{9}{2} \times 8=36$ sq. units
$$ \begin{aligned} & y^{2}-2 y=-x \\\\ & x+y=0 \end{aligned} $$
Area $=\int_{0}^{3}\left(2 y-y^{2}\right)-(-y) d y$
$$ =\int_{0}^{3}\left(3 y-y^{2}\right) d y $$
$$ \begin{aligned} & \left.=\frac{3 y^{2}}{2}-\frac{y^{3}}{3}\right]_{0}^{3} \\\\ & =\frac{27}{2}-9 \\\\ & =\frac{27-18}{2}=\frac{9}{2}=A \end{aligned} $$
$8 A=\frac{9}{2} \times 8=36$ sq. units
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