The area of a surface is the total area of the outer layer of any object. Calculating the surface area of a curved surface is not that easy, like finding the areas of standard shapes like cylinders, spheres or cubes. An extension of the concept of finding the arc length of a curve can be extended to find the surface area over a region, too.

We divide the surface into small subregions and map each subregion onto a tangent plane. Each subregion on the tangent plane has a pair of values for ∆x and ∆y.

For example, if we consider the 𝑖th subregion, then $\Delta x_i\Delta y_i=\Delta A_i$ will be the area of the 𝑖th subregion. Summing up all the approximations to the surface area can be done using two integrals ∬. (Since it is a surface, we have to consider both approximations over 𝑥 and 𝑦 directions).

If 𝑧=𝑓(𝑥,𝑦), where 𝑓𝑥 and 𝑓𝑦 are continuous over a closed, bounded region 𝑅, then the surface area 𝑆 over 𝑅 can be calculated by the following equation.

$$S=\iint_RdS$$ $$=\iint_R\sqrt{1+f_x(x,y)^2+f_y(x,y)^2}dA$$