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First of all, the things that you actually integrate are densities, which are the differential geometric counterparts of measures. No orientation is needed.

A degree $n$ form on an $n$-dimensional manifold is almost a density, but not quite. We need an orientation to associate to the top degree form a density. This is what you ultimately integrate when you integrate a form. For more details see Section 3.4.1 of these notes.

First of all, the things that you actually integrate are densities, which are the differential geometric counterparts of measures. No orientation is needed. On a smooth manifold $M$ with separable topology there is an intrinsic concept of negligible set. A density is then a signed measure $\mu$ such that $|\mu|(B)=0$ for any negligible Borel subset.

A degree $n$ form on an $n$-dimensional manifold is almost a density, but not quite. We need an orientation to associate to the top degree form a density. This is what you ultimately integrate when you integrate a form. For more details see Section 3.4.1 of these notes.

First of all, the things that you actually integrate are densities, which are the differential geometric counterparts of measures. No orientation is needed.

A degree $n$ form on an $n$-dimensional manifold is almost a density, but not quite. We need an orientation to associate to the top degree form a density. This is what you ultimately integrate when you integrate a form. For more details see page 105 of these notes.

First of all, the things that you actually integrate are densities, which are the differential geometric counterparts of measures. No orientation is needed.

A degree $n$ form on an $n$-dimensional manifold is almost a density, but not quite. We need an orientation to associate to the top degree form a density. This is what you ultimately integrate when you integrate a form. For more details see Section 3.4.1 of these notes.

First of all, the things that you actually integrate are densities, which are the differential geometric counterparts of measures. No orientation is needed.

A degree $n$ form on an $n$-dimensional manifold is almost a density, but not quite. We need an orientation associate to the top degree form a density. This is what you ultimately integrate when you integrate a form. For more details see page 105 of these notes.

First of all, the things that you actually integrate are densities, which are the differential geometric counterparts of measures. No orientation is needed.

A degree $n$ form on an $n$-dimensional manifold is almost a density, but not quite. We need an orientation to associate to the top degree form a density. This is what you ultimately integrate when you integrate a form. For more details see page 105 of these notes.