Why does an integral change signs when flipping the boundaries?

Question 1

Let us define a very simple integral:

$f(x) = \int_{a}^{b}{x}$

for $a,b\ge 0$ .

Why do we have the identity $\int_{a}^{b}{x} = -\int_{b}^{a}{x}$ ?

I drew the graphs and thought about it but to me integration, at least in two-dimensions, is just taking the area underneath a curve so why does it matter which direction you take the sum?

Question 2

Here’s another intuitive justification. The obvious graphical intuition says that when $a \leq b \leq c$ , then $\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx$ . If we want this formula to hold for arbitrary $a,b,c$ , then we should be able to take $a=c$ , so that $\int_a^b f(x) dx + \int_b^a f(x) dx = \int_a^a f(x) dx$ . But $\int_a^a f(x) dx = 0$ , so if we want this formula to hold, we need $\int_a^b f(x) dx = -\int_b^a f(x) dx$ .

Why does an integral change signs when flipping the boundaries?

Answer

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