Computing the Surface Area of an Implicitly Defined Surface

In this example, we compute the surface area using the formula for implicit surfaces that we talked about in the previous video of my vector calculus playlist (link below). This paraboloid could be tackled a few ways (parametrically, implicitly or explicitly), but we are going to use the implicit formula thinking of this surface as being above the xy-plane. The double integral that we get actually turns out to be easier to compute in polar coordinates so we make that conversion as well.

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[noahbarrow]

i've watched a multitude of math videos but none are both as illuminating and as digestible as yours. Your enthusiasm for the subject makes learning rather opaque concepts easier!

[garmands]

This channel will be massive, no doubt about it. This is straight to the point!

[geekoutnerd]

😂 I spent 1 week looking through your videos to find the volume bound by an implicit function....learned spherical coordinates and triple integrals in the process, finally I also made a summation formulae to test my results with n cubes. Thank you very much!

[Tsoenyana]

Doc, you are the best. This is impossible.
😂🤣

[joyghosh]

Why gredient of z normal.we learned it to be the Stevest direction... Plz explain sir.

[monu_]

How to find surface area of the solid of intersection of two surfaces ?

[j.o.]

Eyyy, managed to do this one on my own as well after a little thinking. Builds my confidence in mathematics. Question to self: does that mean we can ignore the 1 in z + x^2 + y^2 = 1? Sine we only look at the gradient, that constant cancels out each time, so based on this, I'm quite sure we can do so. Our parameterization also ignores it, so we're happy in that regard. But like, if that isn't 1, but a larger number like 4, then the area is clearly larger, so this doesn't make sense. Ahhh, no it does, because then the range of r would change. Yes, now it makes sense. Nice.

[th.n.]

If i have a wear surface which wear has given way to the material and has created step, material retreat, then i can't describe this wear and

[Festus]

What if we had a z-component function in our Gradient eg. delF= <2x, 2y, 3z>, then the denominator would be a function of z, and the numerator would also have a z-function in it.
Would we want to convert to SPHERICAL coordinates to solve the integral?

[NOR_AZIM]

Hi doc...im thinking about dS conversion to dA...what is going to be if the surface not in k-hat direction say the wall of cylinder....in this case, i wanna solve using cartesian coodinate olny neither paramaterization nor polar/cylindrical coordinate system...thanks

[quicktripgas]

Which math software do you recommend? Maple, Matlab, or Mathematica.

[papadatoss]

What if the we have something like a sphere? What am I going to do with the z component there? Should I add 2 explicit integrals for positive and for negatuve z?

[blandconstant]

I might be having a brain fart(if u know what i mean by that) but when u integrated with respect to theta why didnt u actually integrate? Like if we have integral of 5z and we integrare wrt theta shouldnt it just be 5z*theta?

[consumeentertainment]

Can anyone verify what I am going to say?

F(x, y) = combination of x and y.

The gradients of F lives in 2D plane.

In this case,
F(x, y, z) = constant

The gradients live in 3D space and tangent to the surface because all points because all points of the surface get mapped to 1.

[Karatemaci]

In the u substitution at 6:10 the boundaries of the integral should be 1 and sqrt(5) instead of 0 and 1.