In the first part of example, at time 8:40, when you were integrating between 0,2, you took f(x) only, which equals f(t), why you did not take f(y) also, ? or is it because we are asked only for x, since z= x. Thanks
The method that I presented here is for integrating a given (explicit) function f(x,y) over a given 2-dimensional curve C. For some applications it might be necessary to solve for f(x,y) from an implicit equation!
I don't understand at 10:25, how can a curvy line be put onto a straight surface, isn't that the reason why the map of the earth is so inaccurate. We can't put something curvy on a straight plane.
As a math purist, I'm disappointed that you resorted to "velocity" to convert ds to |r'(t)| dt when this is just a straightforward computation of arc length ...
I find the *r* - *v* - *a* and | *v* | = ds/dt relationships extremely useful for intuitive problem-solving, so I'm always happy to see them put to good use :)
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In the first part of example, at time 8:40, when you were integrating between 0,2, you took f(x) only, which equals f(t), why you did not take f(y) also, ? or is it because we are asked only for x, since z= x. Thanks
See 8:28. We're given that the function is f(x,y) = x.
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Can you explain definite Integration of The greatest integer function?
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See this video for an explanation: ruclips.net/video/5gtf9OuHa9Y/видео.html
Thanks for such an amazing vedio.
Can we say that the explanation given by you was solely for explicit functions 《z=f(x,y)》 ?
The method that I presented here is for integrating a given (explicit) function f(x,y) over a given 2-dimensional curve C. For some applications it might be necessary to solve for f(x,y) from an implicit equation!
what if the curve is a closed curve ?
The integral you get is the magnitude of the vector valued function prime with the limits plugged in?
It's just the magnitud of the curved area
I don't understand at 10:25, how can a curvy line be put onto a straight surface, isn't that the reason why the map of the earth is so inaccurate. We can't put something curvy on a straight plane.
Right, and that's why taking the line integral with dx or dy is DIFFERENT from taking it with respect to ds. You're calculating a different area!
As a math purist, I'm disappointed that you resorted to "velocity" to convert ds to |r'(t)| dt when this is just a straightforward computation of arc length ...
I find the *r* - *v* - *a* and | *v* | = ds/dt relationships extremely useful for intuitive problem-solving, so I'm always happy to see them put to good use :)