Why does an integral change signs when flipping the boundaries?

Let us define a very simple integral:

  • f(x)=bax

for a,b0.

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?

Answer

Here’s another intuitive justification. The obvious graphical intuition says that when abc, 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.

Attribution
Source : Link , Question Author : David South , Answer Author : Gregory J. Puleo

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