wow I thougth I can learn something watching this movie. but u just write down more and more stuff and getting from one line to the next is not explained at all. surprise! o_O
@MistaSmith Good comment.... and Yes, this video assumes that the viewer already knows basic calculus and analytic geometry. So it's not for everybody.
Rather than transforming the variables into polar ones, you can complete the volume integral simply by doing a solid of revolution about the y axis of the graph y=e^(-x^2). Putting in the limits [1,0], watching out for y*log[y] as y tends to zero, results in the volume being pi. So you square root for the single integral and there you go.
kind of nice because this method is contained within the A level syllabus.
@spasman Yes, good comment. [Alternate solution without polar coordinates] @spasman correctly observes: Construction of the volume, double integral, from elementary disks of square radius = log(1/y) reduces to a "do-able" single integral. Evaluation of the limits of integration is accomplished with an interesting limit of Lim [ y-> 0 ] of Log(y^y) = 0.
its a good video, however a little more explanation throughout would be even better, example: at the end of the video 6:50 you didn't mention that the if one were to do e^infinity - e^0 = 1. Although its generally assumed the viewer should know this, its sometimes still nice to mention it, since if you don't, the viewer would have to double check by pausing the video....
The first step: why does G^2 equal the integral dx, and then the integral dy ? Doesn't a quantity squared mean the quantity multiplied by itself? (so that it should be dx and dx)
Yes, I know it can be a confusing point. To understand, first notice that the variable of integration in G does not appear in the answer. G is a number, not a function. That means I have the freedom to pick any variable name without affecting the numerical value of the integral. So "you can't stop me" from picking dx and dy which then allows me to rewrite G^2 as a double integral. I hope this answers your question. ...
Yes, GAMMA(1/2)=SQRT(PI) comes from changing variables in the integral representation for GAMMA(1/2). Changing variables from t to x with t=x^2 dt=2xdx reduces the GAMMA integral to a Gaussian integral. Good point!
wow I thougth I can learn something watching this movie. but u just write down more and more stuff and getting from one line to the next is not explained at all. surprise! o_O
MistaSmith 4 months ago
@MistaSmith Good comment.... and Yes, this video assumes that the viewer already knows basic calculus and analytic geometry. So it's not for everybody.
Mathview 4 months ago
Rather than transforming the variables into polar ones, you can complete the volume integral simply by doing a solid of revolution about the y axis of the graph y=e^(-x^2). Putting in the limits [1,0], watching out for y*log[y] as y tends to zero, results in the volume being pi. So you square root for the single integral and there you go.
kind of nice because this method is contained within the A level syllabus.
spasman 8 months ago
@spasman Yes, good comment. [Alternate solution without polar coordinates] @spasman correctly observes: Construction of the volume, double integral, from elementary disks of square radius = log(1/y) reduces to a "do-able" single integral. Evaluation of the limits of integration is accomplished with an interesting limit of Lim [ y-> 0 ] of Log(y^y) = 0.
Mathview 8 months ago
Awesome!
SevenRiderAirForce 1 year ago
I'm posting some revised versions of these older videos. You may want to view the revised version instead of the original.
Mathview 1 year ago
Great stuff for those ppl who are doing this kind of math as a hobby =)
urbandiscipline88 1 year ago
Yeh who else hates the Gaussian integral?
HumanTargetAus 1 year ago
its a good video, however a little more explanation throughout would be even better, example: at the end of the video 6:50 you didn't mention that the if one were to do e^infinity - e^0 = 1. Although its generally assumed the viewer should know this, its sometimes still nice to mention it, since if you don't, the viewer would have to double check by pausing the video....
chutsu 2 years ago
where did the "r" go cuz du=2r dr (6:02) and then in (6:25) appers jus "2" what did you do to get rid of that "r"?
felozero 2 years ago
Comment removed
greekgod8591 2 years ago
very good!!!!!!!!!!!!
123MATEMATICAS123 2 years ago
Interesting.
higheddy89 3 years ago
Good.
pollardrho06 3 years ago
Thanks,, I plan to do some more with a focus on special functions when I get back to this next month.
Mathview 3 years ago
The first step: why does G^2 equal the integral dx, and then the integral dy ? Doesn't a quantity squared mean the quantity multiplied by itself? (so that it should be dx and dx)
eholmes80 3 years ago
Yes, I know it can be a confusing point. To understand, first notice that the variable of integration in G does not appear in the answer. G is a number, not a function. That means I have the freedom to pick any variable name without affecting the numerical value of the integral. So "you can't stop me" from picking dx and dy which then allows me to rewrite G^2 as a double integral. I hope this answers your question. ...
Mathview 3 years ago
tambien se puede hacer por gamma(x) ya que GAMMA(1/2)=sqrt(pi) es mas facil...
raziel17858 3 years ago
Yes, GAMMA(1/2)=SQRT(PI) comes from changing variables in the integral representation for GAMMA(1/2). Changing variables from t to x with t=x^2 dt=2xdx reduces the GAMMA integral to a Gaussian integral. Good point!
Mathview 3 years ago