Calculating Surface Area Using Parametrization: Tilted Plane Inside Cylinder

  • Thread starter cos(e)
  • Start date
  • Tags
    Surfaces
In summary, the problem involves finding the area of a tilted plane inside a cylinder using parametrization and a double integral. The surface is expressed as r(x,y) and the integral is taken over the region in the x-y plane. There may be a discrepancy in the answers due to a mistake in the given solution.
  • #1
cos(e)
27
0

Homework Statement


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



Homework 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


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


yet the answers have:
Area=integral 0 to 2pi(integral from 0 to 1(sqrt(5)r/2))drd(theta)
=sqrt(5)*pi/2


can someone please help?
 
Physics news on Phys.org
  • #2
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
thanks :)
 

1. What is a parameterised surface?

A parameterised surface is a mathematical representation of a surface in three-dimensional space, where the coordinates of each point on the surface are expressed in terms of one or more parameters. This allows for the easy manipulation and analysis of the surface.

2. What is the purpose of using a parameterised surface?

The purpose of using a parameterised surface is to make it easier to work with and analyze complex surfaces. By expressing the coordinates in terms of parameters, it becomes easier to manipulate and understand the surface's properties, such as curvature and tangent vectors.

3. How is a parameterised surface different from a regular surface?

A regular surface is typically defined by an equation in terms of x, y, and z coordinates, while a parameterised surface is defined by a set of equations in terms of parameters. This allows for more flexibility and ease of manipulation with parameterised surfaces.

4. What are the advantages of using a parameterised surface?

One of the main advantages of using a parameterised surface is that it allows for more efficient and accurate calculations of surface properties, such as curvature and normals. It also makes it easier to visualize and manipulate complex surfaces.

5. How are parameterised surfaces used in real-world applications?

Parameterised surfaces have a wide range of applications in various fields, including computer graphics, engineering, and physics. They are used to model and analyze surfaces in 3D modeling, simulation of physical systems, and design of complex structures, among others.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
984
  • Calculus and Beyond Homework Help
Replies
1
Views
562
  • Calculus and Beyond Homework Help
Replies
4
Views
765
  • Calculus and Beyond Homework Help
Replies
1
Views
416
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
827
Back
Top