When $f$ is integrable, then we say that there exits a partition for a given epsilon such that
$$
U(f,P) - L(f,P) \lt \epsilon \\
\sum_{i=1}^{n} (M_i - m_i) (x_i - x_{i-1}) \lt \epsilon $$
Then, what is actually small in that summation? Is $(M_i - m_i)$ small (that is the difference between sup and inf of $f$ in an interval) or the length of interval $(x_i - x_{I-1})$ is itself small?
$$
U(f,P) - L(f,P) \lt \epsilon \\
\sum_{i=1}^{n} (M_i - m_i) (x_i - x_{i-1}) \lt \epsilon $$
Then, what is actually small in that summation? Is $(M_i - m_i)$ small (that is the difference between sup and inf of $f$ in an interval) or the length of interval $(x_i - x_{I-1})$ is itself small?