TITLE: Curiosity about the area between the arclength and the connected line
QUESTION [5 upvotes]: For example, $f(x)=x^2$
I'm curious if it's possible to find the area, for example, between the function $f(x)$ and the connected line between $f(1)$ and $f(2)$?

REPLY [3 votes]: Call the line connecting $f(1)$ and $f(2)$ $g(x)$:
$$g(x)-f(1)=\frac{f(2)-f(1)}{2-1}(x-1)$$
$$g(x)=3(x-1)+1$$
$$g(x)=3x-2$$
The area you're looking for can then be found by integration:
$$\int_1^2(g(x)-f(x))dx$$
It is $g(x) - f(x)$ rather than $f(x) - g(x)$ because for all $x\in[1,2]$, $g(x)\ge{f(x)}$.