What type of function is $x^2+y^2+6x$?

Question

So, the task is to calculate the volume of body formed when the surface bounded by these functions: $y=x^2/3$ and $x^2+y^2+6x$ rotates around $x$ axis.
Everything involving the actual calculation using definite integrals, finding the limits of integral, etc. is not a problem. The real problem is, I don't understand the function: $x^2+y^2+6x$

I hope not understanding the function won't influence calculating the volume later, but anyway, when I'm given a function, for example, like this one: $y-x$, not like $y=x$ or $y-2x=0$, so without equality, what do they represent and how can I write them in form $y=f(x)$?

Answer 1

Assuming that what is meant is $x^2+y^2+6x = 0$, we can write $x^2+y^2+6x+9 = 9$. The point of doing this is to be able to write

$$ (x+3)^2+y^2 = 9 $$

which shows that this is a circle of radius $3 = \sqrt{9}$, centered at the point $(-3, 0)$.

---

It occurs to me that one other possibility afforded by the problem is that the curve is given by

$$ y = x^2+y^2+6x $$

That's a bit odd, but it too can be worked into the form of a circle:

$$ (x+3)^2+\left(y-\frac{1}{2}\right)^2 = \frac{37}{4} $$

The end result is, of course, not quite as nice.