I was doing some investiations into 'areas under curves' an found by a geometric analogy the following integral identity: $$ \int_0^a f(x)dx =\frac12 \int_0^{f ′ (a)} (a- f^{-1}(f′(x)))^2 dx $$ Now im not sure whether this applies to all classes of functions, but it seems to work for polynominals. I am asking whether there is a field in math about such identities, and what it is called?
And how would one formally prove the identity?