Mathematics - Area Under Curves Question with Solution | TestHub

To find the points of intersection, substitute $x$ from the line equation into the parabola equation.

From the line equation $y=2x-1$, we get $x=\frac{y+1}{2}$.

Substituting this into the parabola equation $y^2=4x-2$:

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

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

$y^2-2y=0$

$y(y-2)=0\Rightarrow y=0,y=2$

The area of the bounded region is given by integrating the difference of the $x$-coordinates from $y=0$ to $y=2$:

$A={\int }_0^2(x_{\text{line}}-x_{\text{parabola}})dy$

$A={\int }_0^2\left(\frac{y+1}{2}-\frac{y^2+2}{4}\right)dy$

$A=\frac{1}{4}{\int }_0^2(2y+2-y^2-2)dy$

$A=\frac{1}{4}{\int }_0^2(2y-y^2)dy$

$A=\frac{1}{4}{\left[y^2-\frac{y^3}{3}\right]}_0^2$

$A=\frac{1}{4}\left(4-\frac{8}{3}\right)$

$A=\frac{1}{4}\left(\frac{4}{3}\right)=\frac{1}{3}$ sq. units.

Answer: $\frac{1}{3}$