Multiple Integrals 13: Evaluation of Polar Double Integrals

[jiramate999]

@NMLP1 Well, I didn't watch it, I saw the introduction and moved on.

[browno26]

2x speed!!!!!!!!!!!!!!

[ebemc]

captions are hilarious

[jiramate99]

@NMLP1 Well, seriously. This is pretty easy stuff. Its almost hilarious.

[Garykelleher123]

@jiramate99 then why u watching video

[NMLP1]

def helped me out. thanks man.

[jiramate99]

it'll be great if your accent can improve!

[NMLP1]

@jiramate99 dont be a noob.

[sodown]

thanks a lot donny you're a life saver!

[bigstas503]

An easier way of considering the geometry of "r.dr.d(theta)" is as follows.

By definition, arc length = r.(theta) .

Hence, d(arc) = r.d(theta) .

The area of the small rectangle equals to the arc length times the small change in r. In the limit, this becomes accurate.

Hence, dA = r.d(theta).dr .

[illusion422]

Another method would be to use the Jacobian, I believe students have already learned partial derivation if they are solving double integrals.

x = r cos(theta) , y = r sin(theta)

J = |del(x)/del(r) del(x)/del(theta)| |del(y)/del(r) del(y)/del(theta)|

This is the determinant of the partial derivatives.

|cos(theta) -r sin(theta)|

|sin(theta) r cos(theta) |

r cos^2(theta) + r sin^2(theta) = r = Jacobian.

So dA becomes r dr d(theta)

[2112dim]

I think there is a theorem of advanced calculus about the Jacobian in coordinants transforms.

Double integral of f(x,ydx.dy)in R=

Double integral of f(g(u,v),h(u,v)*Jacobian[(x,y)­/(u,v)]du.dv in G

[LongShlong125]

oh sh* he just did just that , maybe I'm missing something and it's for the proof....

[LongShlong125]

If someone doesn't understand why the hell are the circle chips ("rectangles") are treated like straight ones , it is because the length in the middle of the rec (where dony picked the point*) is equal to the length of the rectangle if it was bent straight , easy to prove with calculus and geometry , so you can treat it's area as a normal rectangle and multiply by height for the summation.

(I'm on 4:42 so I might be wrong in here if dony outright subtracted two circle sections)