The area of the region bounded by the parabola  $x=2y^2$ and the line x+ y =1 is

$x=2y^2$

$x+y=1$

We get  $1-y=2y^2$

$2y^2+y-1=0$

$\left(y+1\right)\left(2y-1\right)=0$

$y=-1\left(or\right)y=\frac{1}{2}$

Required area  $=\underset{-1}{\overset{\frac{1}{2}}{\int }}\left(x_1-x_2\right)dy$

$=\underset{-1}{\overset{\frac{1}{2}}{\int }}\left(1-y-2y^2\right)dy$

$=\left[\frac{1}{2}-\frac{1}{8}-\frac{1}{12}\right]-\left[-1-\frac{1}{2}+\frac{2}{3}\right]$$=\frac{27}{24}\Rightarrow \frac{9}{8}$