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# Surface integral ex3 part 3

Parametrizing the top surface. Created by Sal Khan.

## Want to join the conversation?

• Now I know that dS = r*√(2)
But at don't we use "r dr dø", instead of just dr dø.
If not, why?
• dS represents an infinitesimally small area of the the surface, S. You're correct that in polar coordinates, we use r dr dθ to represent this area. However, when we calculate surface integrals, the area of the surface element is already encoded in the magnitude of the cross-product |d s / dr x d s / dθ|, where s (r,θ) is the position vector from the origin to the surface. Each of these derivatives represent a side of dS. Recall that the magnitude of the cross-product of two vectors is the area of the parallelogram formed by them.
• I've been wondering if it is alright to just use x and y as variables, since z=1-x, and the boundaries of x is
-1<=x<=1, y is -1<=y<=1
is that correct? Why or why not?
• well if you integrated from -1 to 1 for both x and y you'd get a square base, which is why he uses the cos and sin.
• When parameterizing x and y, why do we say that r is not fixed at 1 for Surface 3? Is it to account for the fact that the plane z=1-x is cutting into our cylinder at a slant? Wouldn't r be variable when leaving the xy-plane and entering the z-dimension only? While on the xy-plane, wouldn't r always equal 1 since it's a unit circle? Sorry, I'm finding the parametrization to be the hardest part of these.
• z is a function of x. so wherever x is defined, z is also defined. "Wouldn't r be variable when leaving the xy-plane and entering the z-dimension only?" So they go together. Don't know if this makes sense to you.
(1 vote)
• Just wondering why is the r not considered 1? isn't the circle have a radius of 1? thanks!
• Keeping the r value at 1 would only let you parameterize the edge of the circle. We want to parameterize the entire area of the circle, and so we need to vary the radius from 0 to 1. This allows us to go to every point inside the circle.
• wasn't there a mistake in the determinant j component?
becuase the order was wrong on the j component ( i think)
etheirway it gives 0 for the j component too
(1 vote)
• What grade do we generally learn this? (•_•)
(1 vote)
• Single variable calculus is generally taught in 12th grade, the last year of highschool, so multivariable calculus is usually taught in the first year of college. Then again, there are plenty of exceptions. I, for one, am in 11th grade. There are plenty of people learning this even younger than me ¯\_(ツ)_/¯. Go at your own pace!
(1 vote)
• What grade is this generaly learned in
(1 vote)
• If you are pursuing engineering, they teach you this in the first semester and maybe in second year when you are learning pure mathematics.
• Are we trying to sum up the values of all the z coordinates on all three parts of the surface? On S1, this results in a zero since it's on the x-y plane?
(1 vote)
• It looks like you are left-handed, yet the little pointer icon is a right hand. Just bugs me a bit!