Why is the rate of change of area under a curve equal to the height of the curve at the second endpoint?

Question

This is the fundamental rule of calculus, and I see the math process and all, but I can't intuitively understand why the statement is true, like why the second endpoint, why doesn't the first matter? How is this intuitive? Any real life examples?

Answer 1

When $x$ changes by $h$, the area from $x$ to $x+h$ is a rectangle with sides $h$ and $f(x)$ with a little triangle (approximately) on top with an area of $h(f(x+h)-f(x))/2 =(h^2/2)(f(x_h)-f(x))/h \approx h^2f'(x)/2 $.

Therefore the area changes by about $hf(x)+h^2f'(x)/2$. Since we consider $h$ to be small, the $h^2f'(x)/2$ term is negligible compared with the $hf(x)$ term, so that is how much the area changes by for small $h$.