Question

Area of the region bounded by the curve $y^2=x$ and the line $x+y=2$, is

• A. $9/2$
• B. $7/2$
• C. $7/6$
• D. $2/3$

Answer 1

The correct option is A $9/2$

$x+y=2x=y^2$

$y^2+y-2=0$

$y^2+2y-y-2=0$

$y(y+2)-1(y+2)=0$

$y=1,-2$

$x=1,4$

Area$={\int }_{y=-2}^{y=1}(2-y-y^2)dx$

$={(2y-\frac{y^2}{2}-\frac{y^3}{3})}_{-2}^1$

$=(2\left(1\right)-\frac{1}{2}-\frac{1}{3})-(2(-2)-\frac{4}{2}-\frac{{(-2)}^3}{3})$

$=(2-\frac{5}{6})-(-4-2+\frac{8}{3})$

$=\frac{7}{6}-(-6+\frac{8}{3})$

$=\frac{7}{6}+6-\frac{8}{3}=\frac{7}{6}+\frac{10}{3}$

$=\frac{27}{6}=\frac{9}{2}$ sq. units