# Write a parameterization of the cone

### Parametrize the portion of the cone

Set up an iterated integral to determine the surface area of this cylinder. These two pieces of information are the parameters u and v. Examples A natural example is a sphere. A one-dimensional curve in space results from a vector function that relies upon one parameter, so a two-dimensional surface naturally involves the use of two parameters. Here r is the radius, the same as the variable in polar coordinates. Here u and v correspond, respectively, to the the spherical coordinates theta and phi. Example 3. A surface is a two dimensional object.

The height is 3, the base radius is 2, and the cone is centered at the origin. Evaluate the iterated integral.

It takes two pieces of information to describe a point on a sphere: the latitude and longitude. Let us compare and contrast the parameterization of a surface with that of a space curve.

### How to parameterize a surface

Two parameters are required to define a point on the surface. Alternatively, a surface can be described in parametric form: where the points u,v lie in some region R of the uv plane. Using the formulas for spherical coordinates we have Here a is a constant, not a variable. These notions will hopefully become clearer as you go through the examples below. We now consider an important example. Figure 3. The height is 3, the base radius is 2, and the cone is centered at the origin. A one-dimensional curve in space results from a vector function that relies upon one parameter, so a two-dimensional surface naturally involves the use of two parameters. Examples A natural example is a sphere. Section 3. In this example, the radius is 2 when the height is 0 and is 0 when the height is 3. Subsection 3. Here u and v correspond, respectively, to the the spherical coordinates theta and phi. The variable r must be eliminated since it is a third parameter. Recall that one way to think about the surface area of a cylinder is to cut the cylinder horizontally and find the perimeter of the resulting cross sectional circle, then multiply by the height.

Each coordinate x,y and z depends only on one parameter, t. Here u and v correspond, respectively, to the the spherical coordinates theta and phi.

These notions will hopefully become clearer as you go through the examples below. Draw the surface defined by your parameterization with appropriate technology. In this example, the radius is 2 when the height is 0 and is 0 when the height is 3.

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