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 $z=f(x,y)$

above a particular region $D$ in $xy$-plane. 

Theorem: 

The area of the surface with equation $z=f(x,y)$, $(x,y)\in D$, where $f_x$ and $f_y$ are continuous is 

$$A(S)=\underset{k,l\to \mathrm{\infty }}{lim}\sum _{i=1}^k\sum _{j=1}^l\mathrm{△}{T}_{ij}=\int {\int }_D\sqrt{f_x^2+f_y^2+1}dA$$

Example 1: Find the surface area of $x+2y+z=4$ that lies above the region $D=\{(x,y)|-1\le x\le 0,0\le y\le 1\}$. 

Exercise 1: Find the surface area of $x+2y-z=1$ that lies above the region $D=\{(x,y)|0\le x\le 1,0\le y\le 2\}$. 

Example 2: Find the surface area of $x-y+2z=4$ that lies inside $x^2+y^2=9$.

Exercise 2: Find the surface area of $3x+y+z=4$ that lies inside $x^2+y^2=4$.

Example 3: Find the surface area of $-x^2-y+2z=4$ that lies above the triangle with vertices $(0,0)$, $(1,0)$ and $(1,2)$.

Exercise 3: Find the surface area of $x-y^2+z=2$ that lies above the triangle with vertices $(0,0)$, $(0,1)$ and $(1,1)$.

Example 4: Find the surface area of $z=xy$ that lies within the cylinder $x^2+y^2=4.$

Exercise 4: Find the surface area of $z=x^2+y^2$ that lies within the cylinder $x^2+y^2=1.$

Example 5: Find the surface area of $z=y^2-x^2$ that lies between $x^2+y^2=1$ and $x^2+y^2=4$.

Exercise 5: Find the surface area of $z=x^2+y^2$ that lies between $x^2+y^2=4$ and $x^2+y^2=9$.

Example 6: Find the surface area of sphere $x^2+y^2+z^2=4$ that lies within $x^2+y^2=2x$.

Group work:

1. Find the surface area of sphere $x^2+y^2+z^2=4$ that lies within cylinder $x^2+y^2=2y$.

2. Find the surface area of sphere $x^2+y^2+z^2=4$ that lies above the plane $z=1$.

3. Find the surface area of sphere $x^2+y^2+z^2=6z$ that lies inside paraboloid $z=x^2+y^2$.