Graphical intuition of why the integral is a linear operator

Question

so I can prove using upper and lower riemansums why the integral is a linear operator:

$$\int a f(x) + b g(x) \,\mathrm{d}x = a\int f(x)\,\mathrm{d}x + b\int g(x)\,\mathrm{d}x$$

And I also understand why it intuitively must be true, as I know that integration and derivation are inverse operations and since the latter is linear the former also has to be.

Graphically I can also understand why a sum of integrals must be the integral of each function $\int (f+g) = \int f + \int g$. However I struggle to understand the graphical implications of the scaling. E.g.

• Is there some graphical intuition why $$ \int a \cdot f(x) \,\mathrm{d}x = a \int f(x) \,\mathrm{d}x$$ holds ?

I tried to graph the above statement for $f(x)=x^2$ on the interval $x \in [0,1]$

However, I can not intuitively say why the green area is twice as big as the green one.

Answer 1

What if one imagines $a f(x)$ as being $a$ copies of $f(x)$ stacked on top of each other?

Then the integral of $a f(x)$ would be $a$ integrals performed “side by side”. With $a$ copies of the area, one can intuit that the cumulative area is $a$ times the original area.