Let $M$ be the manifold, then we know that tangents space has basis $\frac{\partial}{\partial{x_i}}$ and it acts on $g:M \rightarrow \mathbb{R}$ as $\frac{\partial{g}}{\partial{x_i}}$
and cotangent space has basis $dx_i$ and it acts on tangent vectors. Now using this insight, I want to reverese the process as follows:
I first start with cotangent spaces and define it in terms of integral and then difine tangent space as its dual.
Let $f: \mathbb{R} \rightarrow M$ (compare the range and domain with g). Let $\omega$ be a $1$-form i.e. expression of the form $a\;dx + b\;dy$ such that $\omega(f) = \int_f\omega$. The problem with this formulation is I cant extend it to cotangent fields in the following sense:
In the case of the tangent field, $\big(X(f)\big)_p = X_p(f)$ where as this is clearly not true for my definiton of cotangent space.
One thing I notice is that differentiation is a local concept whereas integration makes sense over a range, so the flaw I face in my definition is probably unrepairable. Anyways if possible my aim is to define cotangent space using the concept of integration (just as tangent space was defined using the concept of differentiation/directional derivative).
Bonus Question: In elementary calculus, vaguely speaking we had two important entity; $\frac{d}{dx}$ for differentiation and $dx$ for integration. So is it safe to say that the tangent space and cotangent space is the generalisation of the above two concept respectively for abstract manifold.