It may help to take a step back and look at the basic definitions. The area and centroid are given by
$$ A=\int\\!\\!\\!\int dy~dx=\int y(x)~dx\\\ R_x=\frac{\int\\!\\!\\!\int x~dy~dx}{\int\\!\\!\\!\int dy~dx}=\frac{1}{A}\int x~y(x)~dx\\\ R_y=\frac{\int\\!\\!\\!\int y~dy~dx}{\int\\!\\!\\!\int dy~dx}=\frac{1}{2A}\int y^2(x)~dx\\\ $$
And finally, Pappus's $2^{nd}$ Centroid Theorem states that the volume of a planar area of revolution is the product of the area $A$ and the length of the path traced by its centroid $R$, i.e., $V=2πRA$. Therefore, for rotation about the $y$-axis, we can say that
$$V=2\pi\int_0^{x_{max}} x~y(x)~dx$$
where $x_{max}$ is the point on positive $x$-axis where $y=0$, i.e.
$$x_{max}=\frac{1+\sqrt{1+4ac}}{2a}$$
Thus
$$V=2\pi\int_0^{x_{max}} x~(c+x-ax^2)~dx$$
You should be able to take it from here.