Curiosity about the area between the arclength and the connected line

Question

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 $f(2)$?

Answer 1

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 $x\in[1,2]$, $g(x)\ge{f(x)}$.

Answer 2

Let $g(x)$ be the line connecting the two points from $a$ to $b$, then integrate: $$\int^b_ag(x)-f(x)\ dx$$

In your case, $f(x)=x^2$, the equation of the line connecting $1$ and $2$ would be $g(x)=3x-2$

Thus, the area would be $$\int^2_1(3x-2)-x^2\ dx$$