Depends on the definition of calculus. Without the fundamental theorem (which absolutely requires differentiation), explicitly calculating those areas is unrea

Depends on the definition of calculus. Without the fundamental theorem (which absolutely requires differentiation), explicitly calculating those areas is unreasonably hard (undoable in all but the simplest cases, and in the cases where it is doable, you usually have to be Archimedes to do it). Sure, you could start the course with numerical integration, but this really wouldn't help anything else. (You might be interested, though, to know that your suggestion is essentially how higher math is sometimes done. The natural logarithm function is defined by ln(x)=int_1^x(1/x)dx, which if you look closely doesn't involve exponentials in any way, allowing you to non-circularly define the exponential function as the inverse of the logarithm function, and once you've got the exponential function, the game is on)