Hello,
first of all, the Integrand as a (positive) square-root of a sum of squares is greater or equal Zero. As is pointed out later, I think it can be Zero at only one Point, and that means, you integrate over a positive Integrand, so the integral should be positive and non-Zero.
Look at it from a geometrical Point of view. Your Integrand Looks like the norm of a vector, in fact the norm of the sum of two vectors,
XX = {x, y};
VV = {Cos[phi], Sin[phi]};
And this is a circle of radius 1 centered a the Point { - x, - y }. So your Integrand is in fact the length of a vector from the origin to a Point of that circle. That length is always positive, only if the circle passes through the origin it can be Zero, at exactly one Point. So, the Integrand is (mostly) positive ant the integral non-Zero.
Let’s check it. Define the norm or length of a vector by
norm[a_] := Sqrt[a.a]
and we get
In[11]:= norm[XX + VV]
Out[11]= Sqrt[(x + Cos[phi])^2 + (y + Sin[phi])^2]
exactly your Integrand.
In other words you integrate over the norm[ XX + YY ].
The vector nearest to the origin is XX = { 0, 0 } and as norm[ YY ] is 1 the integral should then become 2 Pi.
Taking into account the inequality
norm[a + b ] <= norm[a] + norm[ b]
and norm [ a ] = constant we arrive at (at least in my opinion)
2 \[Pi] <= integral <= 2 \[Pi] ( 1 + norm [XX] )
Now let us rotate the Frame of reference. Define a Rotation Matrix dd
dd = ( {
{Cos[t], Sin[t]},
{-Sin[t], Cos[t]}
} );
and apply it to the vector in question. Note that this is an orthogonal Transformation and lengths are not changed
In[21]:= vecnew = dd.(XX + VV) // Expand // TrigReduce
Out[21]= {Cos[phi - t] + x Cos[t] + y Sin[t],
y Cos[t] + Sin[phi - t] - x Sin[t]}
We Chose the angle t in a way, that the Point { x, y } is rotated to the x-axes of the new System
In[22]:= vecnew1 =
vecnew /. Cos[t] -> x/Sqrt[x^2 + y^2] /. Sin[t] -> y/Sqrt[x^2 + y^2]
Out[22]= {x^2/Sqrt[x^2 + y^2] + y^2/Sqrt[x^2 + y^2] + Cos[phi - t],
Sin[phi - t]}
Note that this is the same Situation as before, only the Point in question is tranferred to the x-axes of a new coordinate - System. And now we have
j1 = norm[vecnew1] // FullSimplify
Sqrt[1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2] Cos[phi - t]]
But this is of the form Sqrt [ a + b Cos [ w ] ], as the Phase shift doesn’t matter when we integrate over phi from 0 to 2 Pi.
Indeed we get
In[28]:= J1 = \[Integral]j1 \[DifferentialD]phi
Out[28]= (2 Sqrt[1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2] Cos[phi - t]]
EllipticE[(phi - t)/2, (4 Sqrt[x^2 + y^2])/(
1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2])])/Sqrt[(1 + x^2 + y^2 +
2 Sqrt[x^2 + y^2] Cos[phi - t])/(1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2])]
(hmm, interstingly the rotation angle is maintained in the Elliptical thing…) and
In[30]:= T2 = J1 /. phi -> 2 Pi
Out[30]= (2 Sqrt[1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2] Cos[t]]
EllipticE[1/2 (2 \[Pi] - t), (4 Sqrt[x^2 + y^2])/(
1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2])])/Sqrt[(1 + x^2 + y^2 +
2 Sqrt[x^2 + y^2] Cos[t])/(1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2])]
In[31]:= T1 = J1 /. phi -> 0
Out[31]= -((
2 Sqrt[1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2] Cos[t]]
EllipticE[t/2, (4 Sqrt[x^2 + y^2])/(
1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2])])/Sqrt[((
1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2] Cos[t])/(
1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2]))])
and finally
In[34]:= res = T2 - T1 // FullSimplify
Out[34]= (2 Sqrt[
1 + x^2 + y^2 +
2 Sqrt[x^2 + y^2]
Cos[t]] (EllipticE[\[Pi] - t/2, (4 Sqrt[x^2 + y^2])/(
1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2])] +
EllipticE[t/2, (4 Sqrt[x^2 + y^2])/(
1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2])]))/(Sqrt[(
1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2] Cos[t])/(
1 + x^2 + y^2 + 2 Sqrt[x^2 + y^2])])
This should be the result of the Integration. If Mathematica says 0 that is wrong indeed.
Two final checks
In[35]:= res /. x -> 0 /. y -> 0
Out[35]= 2 \[Pi]
ok, And
In[40]:= res /. t -> ArcSin[y/Sqrt[x^2 + y^2]] /. x -> 5. /.
y -> 3.8 // N
Out[40]= 39.7097
compared to
In[41]:= NIntegrate[((Sqrt[(x - Cos[a])^2 + (y - Sin[a])^2]) /.
x -> 5. /. y -> 3.8), {a, 0, 2 Pi}]
Out[41]= 39.7097
Regards
Hans
