In high school in integral course I was told that in order to calculate the area of R in $\int_R {F ds}$ just put $F=1$ and integrate . I don't understand because $\int_R {F ds}$ ( if $R$ is the area of integration and $ds$ is an element of area ) means the integral of $F$ over the area of R .
I don't understand the purpose of $F=1$ because I don't know what is F actually !
It seems to me that I understand the surface integral in wrong way .
Thanks :)
A silly thing you can do in 1 variable calculus is to integrate 1 on an interval to get the length of the interval. Analogously, integrating 1 on a region in the plane gives you the area of the region. When the region is just the region between a positive function $f(x)$ and the x axis, this is equivalent to what you know about integration to find area from one variable calculus. For surface integrals you have the same analogy, it is just that the formulae to do the calculation are more complicated because you have to take into account that the surface is curved and the region you parametrize in is not.
