Why is differential geometry called differential geometry? Why is differential geometry called differential geometry? Why it is not called differential and integral geometry? Isn't integration and finding areas as important as differentiation? Is it the case that  differential equations are more common and natural in mechanics than integral equations so as a result differential geometry as the language of mechanics is more concerned with differentiation?
 A: A few things to keep in mind:


*

*A smooth manifold is defined by using the standard differential calculus and not integration. 

*When one speaks about the geometry of a smooth manifold one is fundamentally concerned with questions regarding curvature, which uses the notion of a Riemannian metric and covariant derivatives on the manifold. These are defined by differential and tensor calculus (by constructing the tangent and tensor bundles) and not a priori by doing integration on the manifold.

*Integration on a manifold is defined for differential forms, which by them selves are defined by differential calculus.
A question: would you call the field "Topology" instead "Topology and continuous maps" just because in topology one often (maybe even most of the time) deals with continuous maps from one space to another? Keep in mind that continuous maps are defined by using the notion of a topology.
Another question: what is a "name"? A "name" should be a term that acts like a basis for the subject in the same sense that a basis for a linear space or a basis for a topology acts: a short description that should span rest of the subject in your mind and by practise.
There is no doubt that integration on smooth manifolds is extremely important, but it seems to me like adding "Integral" to the name of the field is unnecessary. 
A: There are some books covering differential and integral geometry simultaneously. I think the reason is that differential geometry is more common and, like you said, it finds more applications in other branches of science ,mostly theoretical physics, mechanics, etc. On the other hand, if you get into a more serious book on advanced geometry, you can immediately see that integral geometry continues where differential geometry ends.
A: Local differential relationship is the basis  leading to determination and evaluation of global (in the large) immersion whenever it is needed. 
A: The topics of the differential of a map, differential forms, exterior derivative, covariant derivative, Lie derivative, might suggest the subject title.
