Finding line that divides an area into equal halves. My question is simple, but I am not getting the answers for some reason. 
The question is:  Consider the area enclosed between the graph of $y = 1 - x^2 $and the $x$ axis. Which line parallel to the $x$ axis divides the area into two equal parts?
My solutions are as follows,


Which gives answers $k = 1/2$ and two complex number answers, which are all wrong. Where am I making the mistake? 
 A: Another approach would be to integrate with respect to y, so
$\hspace{.3 in}\displaystyle2\int_b^1\sqrt{1-y}\;\;dy=\frac{1}{2}\left(2\int_0^1\sqrt{1-y}\;dy\right)=\left[-\frac{2}{3}(1-y)^{3/2}\right]_0^1=\frac{2}{3}\implies$ 
$\hspace{.3 in}\displaystyle\left[-\frac{2}{3}(1-y)^{3/2}\right]_b^1=\frac{1}{3}\implies\frac{2}{3}\left(1-b\right)^{3/2}=\frac{1}{3}\implies
(1-b)^{3/2}=\frac{1}{2}\implies$
$\hspace{.3 in}\displaystyle1-b=\frac{1}{2^{2/3}}\implies b=1-\frac{1}{2^{2/3}}=1-\frac{1}{\sqrt[3]{4}}$.
A: We work with the right half of the figure. The area of the region is $\int_0^1 (1-x^2)\,dx$, which is $\frac{2}{3}$.
Let our horizontal line meet the right half of the parabola at $x=a$. Then the area below the parabola
 and above the line is $\int_0^a (1-x^2)\,dx-a(1-a^2)$, which simplifies to $\frac{2a^3}{3}$. Set this equal to $\frac{1}{3}$ and solve for $a$. Now we know $a$, so we can find the height of the horizontal.
Remark: We look at a parabola, say $y=x^2$, and a line perpendicular to the axis of the parabola, meeting the parabola at $A$ and $B$. If $w$ is the distance between $A$ and $B$, then the area cut off by the line from the parabola has area proportional to $w^3$. Now if we want to cut this region region into $2$ equal parts by a line parallel to $AB$, meeting the parabola at $C$ and $D$, then we need $\left(\frac{CD}{AB}\right)^3=\frac{1}{2}$. 
If we know $AB$, then $CD$ is determined, and hence so is the distance between the lines $AB$ and $CD$. A more geometric (and more general) version of this was done more than two millenia ago by Archimedes, in his Quadrature of the Parabola.
