Definite integration of continuous, single-variable functions gives you the area between the function and the $x$-axis. This area is a value, not a function.
Indefinite integration results in a function that describes the area in terms of the independent variable $x$. The result, $F(x)$ of an indefinite integral of a primitive function $f$ gives you a function which, given a value $X$, returns the area under $f$ between $0$ and $X$.
Integration doesn't give you an "equation" of higher "dimension". Indefinite integration gives you a function that can be used to compute area; definite integration gives you a single number that describes area. Now, integration of polynomials does in fact result in a polynomial of a higher degree, which has a natural link to dimensionality, but the two should not be conflated.
When you extend to functions of higher order, it becomes more complicated. This is because the notion of the differential becomes a bit more complicated for functions of more than one variable. For single-variable functions, the differential $dx$ is pretty easy to deal with; for functions of two variables, we need to formally describe what we mean by these differentials, such as $dA$, a differential "area" element, or the product $dx\ dy$. There are different, but equivalent, notions of these that you will encounter if you ever study real analysis in detail. I won't go into them now.
It may also surprise you to know that there are many different kinds of integrals. The integral operation is something very deep and complicated. However, they can all be tied back into the same notion of "infinitely summing" over something. It's just that there are many pathways to describe more or less the same thing!