Suppose I have a function $f(x)$, with gradient $\nabla f(x)$. Suppose I approximate the gradient by means of first-order Taylor expansion, obtaining $\nabla h(y)= \nabla f(x) + \nabla f (x)^\top (y-x)$. Integrating this function with respect to $y$ I obtain the function $h(y)$, that is quadratic in $y$. Is this function an approximation of the original function $f(x)$? I.e. is $h \approx f $?
In words, what I am asking is: the integration of the approximation of the gradient of a function, is an approximation of the function itself?