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  • The area (in sq. units) of the region enclosed between the parabola  an

The area (in sq. units) of the region enclosed between the parabola \mathrm{y^{2}=2 x} and the line \mathrm{x+y=4} is

Option: 1

18


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\mathrm{\begin{aligned} &y^{2}=2 x, \quad x+y=4\\ &y^{2}=2(4-y)\\ & y^{2}+2y-8=0\\ &y=-4,2\\ \end{aligned}}

\mathrm{\begin{aligned} &\text { Area }=\int_{-4}^{2}\left[(4-y)-\frac{y^{2}}{2}\right] d y\\ &=\left[4 y-\frac{y^{2}}{2}-\frac{y^{3}}{6}\right]_{-4}^{2}\\ &=\left(8-2-\frac{8}{6}\right)-\left(-16-\frac{16}{2}+\frac{64}{6}\right)\\ &=\frac{28}{6}-\left(-24+\frac{32}{3}\right)\\ &=\frac{14}{3}+24-\frac{32}{3}=18 \end{aligned}}

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