I was watching a khanacademy video on the volume of solids, but I don't quite understand something...
I tried to evaluate the formula at the very end of the video since it's given that $f(x)$ and $g(x)$ both pass through $(0,0)$ and $(c,d)$, but I ended up getting $\frac{1}{4}(f(c)^2 - 2f(c)g(c) + g(c)^2)$ (since f(0)=0 and g(0)=0, there's no need for subtraction),but that just equals $\frac{1}{4}(d^2-2d^2+d^2) = 0$. And that doesn't sound right... where did I go wrong?
You've got to integrate. He says that the volume is $$\int_0^c \frac 14 (f(x)-g(x))^2 dx $$ which is not evaluatable because you can't evaluate the integral of any given function squared in the case of $$\int f^2-2fg+g^2 dx $$. If $f(x)=g(x)=x$, the integral of $f^2$ would be $1/3 f^3$. But if $f(x)=e^x$, the integral of $f^2$ would most definitely not be $1/3 f^3$. There is no general formula for this integral, you just have to evaluate on a case by case basis.