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# Example of calculating a surface integral part 1

Example of calculating a surface integral part 1. Created by Sal Khan.

## Want to join the conversation?

• How to decide the order of the cross product? i.e is it (r sub s) x (r sub t) OR (r sub t) x (r sub s)?
• With cross products, the only that changes depending on the order is the sign of the result. But here we're taking the absolute value of the result anyway, so it doesn't matter in what order you arrange the vectors in the cross product.
• Hey everyone! I wanted to share a useful tip for understanding parametrization. One way to think about it is as a change of variables. Instead of directly dealing with the surface itself, we introduce parameters (typically u and v) to describe the points on the surface. It's like a transformation that simplifies things. This change of variables allows us to switch to a parameter space, which often makes calculations easier. Interestingly, parametrization is connected to linear transformations and the Jacobian determinant, which further enhances its significance. So, by viewing parametrization as a change of variables, we can gain a deeper understanding of its concept and apply it effectively in various calculations.
• In the video where he parametrized the torus, I arrived at r(s,t) = (b+a*coss)cost î + (b+a*coss)sint ^j + a*sins ^k, thought it was fine because it is radially simetric right?
But now the question rises again, is there any difference? Still don't feel completety sure about parametrization =/
• It is fine. The only difference is the definition of the parameter t.
• Theoretically, could the torus be parameterized in one variable using a Hilbert curve, or other space-filling curve that goes "through" (idk if that's the right terminology) the "s" and "t" square?
(1 vote)
• It could, but then you would need an uncountably infinite sum of different integrals.
(1 vote)
• Can you refer me another helpful video
(1 vote)
• How do you exactly remember the steps to take a surface integral? It seems very hairy and complicated
(1 vote)
• Sal. How do we find the area of a surface of revolution? I think it would be similar to this, but I haven't been able to find a video more closely related to it.
(1 vote)
• I searched also, and I couldn't find a video on KA. The way to find the surface area (SA) is to build on the formula for finding arc length and also the ideas for finding the volume of a solid of revolution. The idea (as with almost ALL integration concepts) is that we will slice the object into many thin slices, and then add up (integrate) an expression for the SA of each slice. When we do the VOLUME of a solid of revolution, we just use discs or washers, since the error from "chopping off the corners" of the actual slices turns out to vanish. But for SA, it does matter that the surface is "diagonal", so we need to understand that the SA of the edge of one slice is the ARC LENGTH of the function times the circumference of the slice. That is, if the surface is formed by revolving a function f(x) around the x-axis, then:
SA = ∫(circumference of slice)*(arc length of edge of slice)
SA = ∫(2π)*f(x)*sqrt(1 + (f'(x))²)dx

If you are not sure why the sqrt(1 + (f'(x))²)dx part gives the arc length of a small piece of curve, you can watch this KA vid:

And if you want to see an example of SA of a surface of revolution, there are several on youtube, such as this: