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?
$\endgroup$ 61 Answer
$\begingroup$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.
