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Let $M$ be a differentiable manifold of dimension $n$ and $\alpha$ a $n$-form defined in $M$. And we would like to make sense of the expression $\int_M \alpha$. We know that a $n$-form gives a reference of volume in each tangent space of the manifold. In these context, if we split up the manifold in a lot of little pieces, each piece seems "straight" in the tangent space and the form $\alpha$ would compute some value in each piece. Then the integral comes to be the limit of these sums when the partition becomes infinitely fine

But this is still quite vague... It seems that curves need to be directed, with initial point and final point. And with the surface... in which order are the vectors plugged in the 2-form $\beta$? Well, it happens that for the integral to be well-defined, manifolds need to be oriented, as happened with the homology