Re: double integral problem

In article <e4o8ki$hs7$1@xxxxxxxxxxxxxxxxxxxxxxxx>,
"Julian V. Noble" <jvn@xxxxxxxxxxxx> wrote:

Helmut Jarausch wrote:
xieshufu@xxxxxxxxx wrote:
Hi everyone,

I have trouble in solving a double integral. Could anyone help me?

\int \int_{D} (sqrt(x^{2}+y^{2})) dxdy
the integral area D is : (x-d)^{2}+y^{2}=r^{2}

I can't get the simplist expression
of the double integral. Can anyone give me a hand?

OK that's probably a homework.
Just one hint, what about polar coordinates?


Here's another hint: where is the origin?

An interesting homework problem then -- because there does not appear to
be a simple closed-form expression for the result. I assume you are
expecting that the OP has studied elliptic functions, because the answer
is

(2/9) (d + r) ((d^2 + 7 r^2) EllipticE[(4 d r)/(d + r)^2] -
(d - r)^2 EllipticK[(4 d r)/(d + r)^2])

which is not completely trivial. This can also be expressed in terms of
Gaussian hypergeometric functions.

As a check, when d = 0, the integral correctly reduces to 2 Pi r^3 / 3.

Cheers,
Paul

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