While reading the mathematical introduction to a physics textbook, I came across a problem that asserted that: $$ \int dA$$ (where A is a plane surface) is a vector with a direction perpendicular to a surface A. In my studies of vector calculus, I remember learning that the integration of a vector (unless its a vector valued function) is not possible. However, now I realize that the above equation should have a direction as the area of A does indeed have a direction.
My question is why don't all integrals give vector results? The differentials all appear to be vectors (as shown above) and thus shouldn't all integrals give vectors?
There is an abuse of notation. In general $\int d\vec A$ is a vector quantity and $\int dA$ is a scalar. Some authors use $dA$ as a vector quantity, but you should be able to get this information from the context.