The Fundamental Theorem of Calculus and Accumulation Functions AB & BC

With the upper limit free to move, $g(x) = \int_{a}^{x} f(t)\,dt$ is a function of $x$: the signed area collected from $a$ to $x$, so $g(a) = 0$ and $g$ falls wherever $f$ is below the axis. The letter $t$ inside is a dummy and never meets the $x$ outside. The Fundamental Theorem of Calculus, Part 1, says that for continuous $f$ this $g$ is differentiable with $g'(x) = f(x)$: pushing the finish line forward by $h$ adds a strip of width $h$ and height about $f(x)$. The starting point $a$ does not appear in the conclusion, because moving it shifts $g$ by a constant.

The marks are lost when the limit is not simply $x$. A composite upper limit brings the chain rule, $\frac{d}{dx}\int_{a}^{u(x)} f(t)\,dt = f(u(x))\,u'(x)$, so the height is read at the finish line and the width is how fast the finish line travels. A variable lower limit flips the sign, giving $-f(v(x))\,v'(x)$, since advancing it removes area. When both limits move, the two contributions subtract, and a single-term answer is the signature of a moving edge that went unnoticed.