Integration of implicit functions

Question

As we know an implicit function $f(x,y)$ is an expression containing $x$ and $y$ such that none of them can be seperated out.
So my question is how can we integrate or simply find area under an implicit function from $x=a$ to $x=b$ as we cannot seperate $x$ or $y$
Is there some special techniques i would love to know it because it seems to be very difficult as per my knowledge.
I really have a doubt so please give nice answers.

Answer 1

There is a potential possibility to calculate all implicit derivatives in the some points. This allows to build and to integrate Taylor series. And if obtained series converges absolutely, then its sum gives required result.

Answer 2

For a given interval $[a,b]\subseteq\mathbb{R}^n$, let a function being defined in implicit form by the following equation $$ f(x,y)=0\quad x\in [a,b]\subseteq\mathbb{R}, y\in\mathbb{R},\tag{1}\label{1} $$ where, obviously, for any given $x\in [a,b]$ there exist only one $y$ which satisfies it. To my knowledge, the only general way to integrate a function given in this form is to consider \eqref{1} as the defining equation of a set in $\mathbb{R}^2$ and measure this set. Precisely, let $$ S_f=\left\{(x,y_o)\in\mathbb{R}^2|x\in D, (y\geq y_o\geq 0\text{ if }y\geq 0)\vee(0> y_0\geq y\text{ if }y<0)\right\} $$ Then the integral of the function defined by \eqref{1} on $[a,b]$ is given by $$ \int\limits_{S_f}\!\mathrm{d}x\mathrm{d}y\tag{2}\label{2} $$ This coincides with the naive definition of the integral of a function as the "area of the surface under the graph of the function itself". As implicitly suggested by Teh Rod, by constructing a map $\varphi:D\to S_f$ from a wisely chosen "nice" domain $D\subseteq\mathbb{R}^2$ the integral \eqref{2} the following equation can be obtained $$ \int\limits_{S_f}\!\mathrm{d}x\mathrm{d}y=\int\limits_D\!|J(\varphi)|\mathrm{d}u\mathrm{d}v,\tag{3}\label{3} $$ where $|J(\varphi)|$ is the Jacobian determinant of the map $\varphi$. As a final note, since we have $\mathbb{R}^2\equiv\mathbb{C}$, then $\varphi$ can be chosen to be a conformal map: this, for particular forms of the equation \eqref{1} and thus of $S_f$, can possibly lead to explicitly integrable right sides of equation \eqref{3}.