Surface Area

It is actually another application of the double integral. It is like the arc length to the single integral, we are also finding a property of the boundary.

Firstly recall the formula of the arc length, we have ${\displaystyle A(t)={\int }_m^n\sqrt{t^{\prime }(x{)}^2+t^{\prime }(y{)}^2}dt}$.

The integrand is the parameter equation of the function. We also use the idea of parameterizing here.

For ${\displaystyle z=f(x,y)}$, we use ${\displaystyle x,y}$ to represent ${\displaystyle z}$, and ${\displaystyle f_x,f_y}$ to represent ${\displaystyle f_z}$.

By dividing the surface area into many small parallelograms and finding their Riemann Sum, we have ${\displaystyle A(G)=\underset{S}{\iint }\sqrt{f_x^2+f_y^2+1}dA}$.