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I just learn something about finding the area enclosed by a function with integral. If there is a function $f(x)$, the integral within $[a,b]$ will the the area that the area enclosed by $f(x)$ in $a$ and $b$ and the x axis. Now, if you have two curves, $f(x)$ and $g(x)$, if they intersect at $x=a$ and $x=b$, to find the area that enclosed by those two curves, we could integrate $f(x)$ in $[a,b]$ and that for $g(x)$ in $[a,b]$. Use the big area to subtract the small area to get the total area enclosed by those two curves. Now if I have a product of two curves $f(x)g(x)$ and I am going to integrate the product in some range, is there any geometric significance like area I could define for that integral? I may be wrong but I just have a feeling that the integral of the product is the area enclosed for those two curves?

user1285419
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    Nope. The difference would be the area between the curves. I don't know of any simple geometric meaning of the product. –  Oct 03 '16 at 23:46
  • Integrals are linear operators. For example: $$∫f(x) + g(x) dx = ∫f(x) dx + ∫g(x)dx$$ Since integrals can be applied through terms, you can take the integral of the difference of the functions, and end up with the area between curves, since the process is the same as finding each area individually. However, you can't do that when you multiply two functions. When it comes to the product, I doubt any good, useful relation exists. – Kaynex Oct 03 '16 at 23:50
  • In fact, the geometric interpretation of integrating the product of two curves exists only if one is the probability density function. – Alex Oct 04 '16 at 00:18
  • @Alex I'd love to see the proof of your theorem about the existence of a geometric interpretation. ;p –  Oct 04 '16 at 00:44
  • It goes approximately like this. If the variable $x$ represents the independent variable, and $f(x)$ is the function that says what you are looking for, and $g(x)$ is the function that says how often $x$ will occur, then the integral of $f(x)g(x)$ will tell you the expected value for $x$. If you need more info, I'll have to post an answer. – Alex Oct 04 '16 at 00:53
  • @Alex I know how probability densities work, I meant that the "only if" part of your claim is unjustifiable because "geometric interpretation" is a mental process and thus not something amenable to mathematical statements. I.e. it was a playful jab at your wording. :-) –  Oct 04 '16 at 02:42

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We can consider $f(x)g(x)$ to be the area of a rectangle that varies as $x$ does.

Then $\int f(x)g(x) dx$ is the volume generated out as this varying rectangle is produced. Provided that $f(x)>0$ and $g(x)>0$, there will be no problem with this geometric meaning.

enter image description here

tomi
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  • The volume could fall into two quadrants. One represents the positive, and the other represents the negative. The two could cancel out each other. – zzzgoo Oct 05 '23 at 05:51