Not sure the title has the correct terminology but I'd like to evaluate the following definite integral:


Does this require the same approach as


effectively returning the additive inverse?

Or is there some kind of voodoo that must be done, such as perhaps using $[-∞, ∞]$ as limits of integration and then subtracting using $[1,5]$?

Also, what does it mean? Does it represent an area as well?


If you swap the bounds, the area will be exactly the same magnitude, but of the opposite sign.

If you graph $y=x^2$, in both cases you are still taking the area under the curve, just in different directions.

For example:


= $[\frac{1}{3}x^3]_5^1$ =

= $\frac{1}{3}(1)^3 -\frac{1}{3}(5)^3$

Which yields a negative answer

Compare this to

$$\int_1^5{x^2}dx$$ = $[\frac{1}{3}x^3]_1^5$ =

= $\frac{1}{3}(5)^3 -\frac{1}{3}(1)^3$

Which gives a positive answer

Further note that $\mid \frac{1}{3}(1)^3 -\frac{1}{3}(5)^3\mid = \mid \frac{1}{3}(5)^3 -\frac{1}{3}(1)^3 \mid$

This is known as the physical area, compared to a negative area which only makes sense mathematically as a number, since you can't really have a "negative area".


We usually define $$\int_a^b f(x)\,\mathrm{d} x = -\int_b^a f(x)\,\mathrm{d}x.$$


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