TITLE: Why is the integral defined as the limit of the sum $\int_a^b f(x) dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x$?
QUESTION [7 upvotes]: I am failing to understand why the integral is defined as:
$$\int_a^b f(x) dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x$$
instead of:
$$\int_a^b f(x)dx=\sum_{i=1}^\infty f(x_i^*)\Delta x$$

Is the former just popular preference or is there something I am not conceptually understanding here?

REPLY [2 votes]: Another point not previously made: Using an infinite sum doesn't actually change anything because an infinite sum is defined as a limit of finite sums.