Calculate the region area of the $xy$-plane limited by curves $C_1$ and $C_2$ with $|x|\leq4$, knowing that $C_1$ passes through $(1, 1)$, $C_2$ passes through $(1, -1)$ and both curves belong to the orthogonal family a $x^2+2y^2=K$, with $K$ arbitrary constant.
I did this:
$$\begin{matrix} \mathfrak F:&2x+4yy'=0&(1)\\\\ \mathfrak F^\perp:&2x-4\dfrac y{y'}=0&(2)\\\\ \mathfrak F^\perp:&x=2\dfrac y{y'}&(3)\\\\ \mathfrak F^\perp:&\dfrac{y'}y=\dfrac2x&(4)\\\\ \mathfrak F^\perp:&\displaystyle\int\dfrac{\text dy}y=2\displaystyle\int\dfrac{\text dx}x&(5)\\\\ \mathfrak F^\perp:&\ln|y|=\ln(x^2)+C&(6)\\\\ \mathfrak F^\perp:&y=kx^2.&(7) \end{matrix}$$
As we have $y(1)=1$ for $C_1$ and $y(1)=-1$ for $C_2$ then $$\begin{matrix} C_1:&y=x^2\\ C_2:&y=-x^2,\\ \end{matrix}$$ so the integral for $|x|\leq4$ becomes $$\text{Area}=\dfrac{256}3,$$ according to WolframAlpha.
Is this correct? By the way I am not sure if the first steps are correct, because when I derive in the first equation $K$ dissapears (I know this is a constant, but sometimes we need to replace $x^2+2y^2=K$ inside the $\mathfrak F$, and this time that did not happen).
Any help?
Thank you!