Is calculus 1 the same as calculus ab?

Short answer: They're similar, but Calculus AB covers a bit more (and goes more in-depth) than Calculus 1.

Long answer:

The AP Calculus AB test covers differentiation (taking derivatives) and early integration (taking antiderivatives), including the concept/applications of an integral and integration by substitution. In college, Calculus 1 covers mostly differentiation and Calculus 2 covers additional strategies for integration and series.

I like to think of it like this:

A = Differentiation

B = Integration

C = Series

So Calculus AB covers differentiation and integration and Calculus BC covers integration and series.

College is more like:

Calc 1 = A

Calc 2 = B&C

Of course, this depends on how much you cover in high school and college.