Generalization of line and surface integrals

I think it's a well-known question, but probably I don't know the correct terminology to find it on the web.

I know that classical integrals over and interval generalize in line integral: so we can integrate over a curve, but in n-dimension. Line integrals split in two: for scalar field and for vector field.

I haven't studied it yet, but I found that double integrals generalize in surface integrals and these also split in two for scalar and vector field. But I have also seen that these use cross product, so they generalize double integral just in 3-dimension.

My question is: if I have an m-dimensional domain in a n-dimensional space, how can I integrate scalar and vector fields over it?

I suppose (this is what I found) that the generalization of an integral of a vector field is an integral of a differential form. But what is the generalization of an integral of a scalar field?

For example, if I want to calculate the surface area of a n-ball, what integral I have to write down? (I already know its value, I am just looking for the writing)

Last: which books can I read on this?