Let us define a very simple integral:
Why do we have the identity ∫bax=−∫abx?
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?
Here’s another intuitive justification. The obvious graphical intuition says that when a≤b≤c, then ∫baf(x)dx+∫cbf(x)dx=∫caf(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 ∫baf(x)dx+∫abf(x)dx=∫aaf(x)dx. But ∫aaf(x)dx=0, so if we want this formula to hold, we need ∫baf(x)dx=−∫abf(x)dx.