Section 4.9 Surface Area
4.9 Surface Area
This section is another application for the integrations. We can use double integration to find the surface area of a function [latex]z=f(x,y)[/latex]
above a particular region [latex]D[/latex] in [latex]xy[/latex]-plane.
Theorem:
The area of the surface with equation [latex]z=f(x,y)[/latex], [latex](x,y)\in D[/latex], where [latex]f_{x}[/latex] and [latex]f_{y}[/latex] are continuous is
\[A(S)=\lim_{k,l\rightarrow\infty}\sum_{i=1}^{k}\sum_{j=1}^{l}\triangle T_{ij}=\int\int_{D}\sqrt{f_{x}^{2}+f_{y}^{2}+1}dA\]
Example 1: Find the surface area of [latex]x+2y+z=4[/latex] that lies above the region [latex]D=\{(x,y)|-1\leq x\leq0,0\leq y\leq1\}[/latex].
Exercise 1: Find the surface area of [latex]x+2y-z=1[/latex] that lies above the region [latex]D=\{(x,y)|0\leq x\leq1,0\leq y\leq2\}[/latex].
Example 2: Find the surface area of [latex]x-y+2z=4[/latex] that lies inside [latex]x^{2}+y^{2}=9[/latex].
Exercise 2: Find the surface area of [latex]3x+y+z=4[/latex] that lies inside [latex]x^{2}+y^{2}=4[/latex].
Example 3: Find the surface area of [latex]-x^{2}-y+2z=4[/latex] that lies above the triangle with vertices [latex](0,0)[/latex], [latex](1,0)[/latex] and [latex](1,2)[/latex].
Exercise 3: Find the surface area of [latex]x-y^{2}+z=2[/latex] that lies above the triangle with vertices [latex](0,0)[/latex], [latex](0,1)[/latex] and [latex](1,1)[/latex].
Example 4: Find the surface area of [latex]z=xy[/latex] that lies within the cylinder [latex]x^{2}+y^{2}=4.[/latex]
Exercise 4: Find the surface area of [latex]z=x^{2}+y^{2}[/latex] that lies within the cylinder [latex]x^{2}+y^{2}=1.[/latex]
Example 5: Find the surface area of [latex]z=y^{2}-x^{2}[/latex] that lies between [latex]x^{2}+y^{2}=1[/latex] and [latex]x^{2}+y^{2}=4[/latex].
Exercise 5: Find the surface area of [latex]z=x^{2}+y^{2}[/latex] that lies between [latex]x^{2}+y^{2}=4[/latex] and [latex]x^{2}+y^{2}=9[/latex].
Example 6: Find the surface area of sphere [latex]x^{2}+y^{2}+z^{2}=4[/latex] that lies within [latex]x^{2}+y^{2}=2x[/latex].
Group work:
1. Find the surface area of sphere [latex]x^{2}+y^{2}+z^{2}=4[/latex] that lies within cylinder [latex]x^{2}+y^{2}=2y[/latex].
2. Find the surface area of sphere [latex]x^{2}+y^{2}+z^{2}=4[/latex] that lies above the plane [latex]z=1[/latex].
3. Find the surface area of sphere [latex]x^{2}+y^{2}+z^{2}=6z[/latex] that lies inside paraboloid [latex]z=x^{2}+y^{2}[/latex].