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# Surface integral ex2 part 1

Parametrizing a surface that can be explictly made a function of x and y. Created by Sal Khan.

## Want to join the conversation?

• Why does y^2 become v not v^2?
• at Sal writes re-writes the surface as z = x + y². At , for his parameterization he just makes the substitution: x=u and y=v. Plug that into the surface function: z = u+ v².

As a position vector valued function that gives you: r(u,v) = (u)i + (v)j + (u+v²)k.
• i didn't get why y was required? all the ds s should give me the area of the surface
• Because you are not calculating the area of the surface in this example. You are calculating the mass of the surface and "y" is the "mass per area", hence "ydS" is the "mass of an infinitesimal surface". Summing all these up, you get the mass of the entire object, which is the surface depicted in the video.
• How do you know how to parametrize a surface? In this video, Sal simply sets x and y directly to their own parameters, whereas in the previous examples, he has to take the sin's and cos's of the parameters to describe x, y, and z. In other words, how do you know when to manipulate the parameters versus simply setting them equal to x, y, or z? Is it just practice/intuition?
• After hours of research, I still confuse about how to choose parameters for parameterization, but I want to share what I though could be correct.
To me, parameterization is a transformation about the inputs of the function from the original one to the new one.
e.g x,y,z as the original inputs, r,θ,Φ as the new inputs.
Or, we could reduce the amount of inputs, e.g: x,y,z -> u,v.
The critical difference is about what you are chasing for, in this video, is the surface area, 2 variables is a reasonable choice because the area formula for infinitesimal area only require 2 arguments, not 3.
(1 vote)
• Why does he use "u" and "v"? Why not stick with "S" and "t"?
• s and t are traditionally used to represent distance and time, Sal mentioned mass and area for this example so he chose a different but still commonly used pair of letters.
• If we can take x and y as parameters then why have we been parameterizing all surfaces upto now like the torus and the unit sphere?
• Why do we take the integral of y?