Computing the Surface Area of a surface parametrically // Example 1 // Vector Calculus

In this video we compute the surface area of the portion of a plane that lies within a cylinder. We've previously derived (see vector calculus playlist below) a formula to compute surface area for a parameterized surface, so our first ask is to find an appropriate parameterization of the plane. The key is that because the boundary is this cylinder, it makes sense to use the polar r and theta parameters so that our parameterization respects the boundary and will make the future integral easy to write down.

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[jamyllecarter]

@Dr. Trefor - An excellent video with a small typo. You start computing surface area, but at 5:58 you write it as arc length. Thanks for educating us all!!

[irenepadre]

i love that he's filming this comfortably * his knees were showing. the explanation is hands down always the best. thank you doc!

[arsalansyed]

you are a blessing to humanity! Thank you Dr. Trefor!

[leebercohen]

These lectures are wonderful. Hopefully you will have a future courses on complex variables and differential geometry. Thank you for your lucid explanations and enthusiasm. Leeber

[nott_applicable]

something interesting to note about this surface area is that you could have also found it using a clever trick.

Since our intersection is an ellipse one might think to calculate the area of a certain circle and stretch it. Indeed here if you took the area of the cylinder (projected flat, so more like the area of the circle with radius 2) (pi r^2 = 4pi) and multiplied it by the "stretch factor" you would have gotten the same answer. How do we get sqrt(2) as stretch? It actually makes quite a bit of sense.

imagine having a line of length 1 from (0, 0) to (1, 0) and now moving the endpoint of this line such that the new endpoint is (1, 1), this line now has length sqrt(2) and thus has grown with factor sqrt(2). Now imagine the same with our circle, lay it flat, and rotate it 45 degrees while maintaining the endpoints. It will also stretch by sqrt(2) giving us our final answer of 4pi*sqrt(2) as well.

Obviously very specific to this case but still a nice trick I wanted to share. Regardless, great video as always.

[Thenit]

Your note in the end about the jacobian, r dr is gold!

[j.o.]

Was happy I managed to do this on my own. Question to self: How would I parameterize if it wasn't the obvious z = -x (which is explicit with regards to z)? I would certainly still use polar coordinates cause it makes sense in the context. The difference would be that the plane is an ODE with 2 or more solutions for z, perhaps. In which case, there might be multiple, wiggly planes cross through the cylinder. Then it makes sense to solve for each value of z, then add them up. I know some ODE's don't have an explicit solution and you need to use numerical methods to solve those, like the Runge Kutta method. Either way, I think that's beyond the scope of what's currently going on here, so I'm satisfied with my solution to more than one solution for z

[whoviandoctor]

thank you funny bearded man for saving my desk from the destruction of my fists after getting the same problem wrong 20 times.

[diegoalejandrotellezmartin]

I just don't understand who is that only person who dislike the video .

[gabitheancient]

I don't understand, why can you substitute in z = -x in the parametrization of the cylinder? how wouldn't that make the possibility you find the area of another part of the cylinder? I'm very confused
what subing in z = -x even mean geometrically?
EDIT: ok I kinda have a visualization in my head rn, like for each point in the circle of the xy plane you're associating a point z that's equal to the -the x-coordinate of that point, which in polar coordinates (the parametrization in question) is rcos(theta)

[jamwheeler]

kind of off topic but i wanted another professor's opinion. at the university I'm attending for mech. eng. they teach multivariable and vector calculous in 1 single calculous 3 course. is this normal or even reasonable? i have maintained a 100 in cal 1 and 2 but this cal 3 is looking like I'm going to end up in the upper 90s and as amazing as my professor is he doesn't have the time to go into great detail about this course so ive been supplementing his class with lots of reading and youtube watching. im one of those people that doesn't just want to be able to plug numbers into a formula but i enjoy understanding exactly whats going on and why. (which is why i greatly enjoy your content, thanks so much for what you do). but any who i was just curious as to what your thoughts were on teaching these together. I love you content and am constantly thinking about switching majors or getting another degree in some type of math. thanks again.

[devashishshah]

Sir, if we are given a surface area as a complicated double integral w.r.t. x and y then we will have to multiply by the Jacobian if we want to reparameterize say to polar coordinates, right?

[roshanpradeep]

Why didn't we consider radius(r) to be constant here?