The original function **cannot be recovered** from the result of a definite integral unless it is known that $f(x)$ is a constant.
## N-dimensional integrals
Much like how $dx$ represents an infinitely small line, $dx\cdot dy$ represents an infinitely small rectangle. This means that the surface area of an object can be expressed as:
$$dS=dx\cdot dy$$
Therefore, the area of a function can be expressed as:
$$S=\int^x_0\int^y_0 dy\ dx$$
where $y$ is usually equal to $f(x)$, changing on each iteration.
!!! example
The area of a circle can be expressed as $y=\pm\sqrt{r^2-x^2}$. This can be reduced to $y=2\sqrt{r^2-x^2}$ because of the symmetry of the equation.
$$
\begin{align*}
A&=\int^r_0\int^{\sqrt{r^2-x^2}}_0 dy\ dx \\
&=\int^r_0\sqrt{r^2-x^2}\ dx
\end{align*}
$$
!!! warning
Similar to parentheses, the correct integral squiggly must be paired with the correct differential element.
