I know the definite integral of a function can be thought of as the area between the function and y=0 and between the lower and upper bounds of integration, so long as the function is positive in that region.(adsbygoogle = window.adsbygoogle || []).push({});

However, I also know that:

[tex]\int_{a}^{b} f(x) ~dx = -\int_{b}^{a} f(x) ~dx[/tex]

And I find it hard to reconcile these two things. It would be nice to think of the integral as a process where you're adding up a whole bunch of little slices between the function and y=0 from the lower bound to the upper bound, but under this rationalization, you'd be adding up exactly the same thing whether you add them from left to right or right to left. The "area under the curve" would remain exactly the same even if you mirrored the whole thing horizontally.

How do you think of this? Do you have a way of looking at it that neatly accounts for the fact that inverting the bounds causes the sign to change?

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# Interpretation of integral as area?

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