Question

proof of fundamental theorem of calculus(in 18.01's text book) assumes that A(x) = area under a curve between a and x can be defined.

Can we assume it?

Answer 1

Yes, because we assume a function is a continuous function on a closed interval [a, b]. If F'(x) = f(x) then \[\int\limits_{a}^{b} f(x)dx = F(a) - F(b)\]

This works because we know our function f is continuous on the interval [a, b] and we defined F'(x) = f(x).

What if we have a situation such as \[\int\limits_{1}^{\infty} f(x)dx = \lim_{b \rightarrow \infty} \int\limits_{1}^{b} f(x) dx\]

Obviously the Riemann integral is not well defined so you would have to test for convergence, and you would want to ask yourself if the limit exists and if it can be computed.

Hope it helps.