# Parameterised surfaces

1. Jun 10, 2009

### cos(e)

1. The problem statement, all variables and given/known data
Use parametrization to express the area of the surface as a double integral. tilted plane inside cylinder, the portion of the plane y+2z=2 inside the cylinder x^2+y^2=1

2. Relevant equations
the area of a smooth surface
r(u,v)=f(u,v)i+g(u,v)j+h(u,v)k a<=u<=b c<=v<=d
is
A=integral from c to d ( integral from a to b(|r subu X r subv|))dudv

3. The attempt at a solution
x=x z=z y=2-2z
r(x,z)=xi+(2-2z)j+zk
r subx=i
r subz=-2j+k
r subx X r subz=| i j k |=-j-2k
| 1 0 0 |
| 0 -2 1 |

|r subx X r subz |=sqrt(5)

Area=integral 0 to 2pi(integral from 0 to 1(sqrt(5)r))drd(theta)
=sqrt(5)*pi

Area=integral 0 to 2pi(integral from 0 to 1(sqrt(5)r/2))drd(theta)
=sqrt(5)*pi/2

2. Jun 10, 2009

### Dick

You are integrating over the region in the x-y plane right? So you want r(x,y). And you want to integrate |r subx X r suby|. Express the surface in terms of x and y coordinates. Not x and z.

3. Jun 10, 2009

thanks :)