Define

x = r Cos[t];
y = r Sin[t];

Then

x^2 + y^2 // Simplify

r^2

So your r is equal to 6.

Your constraints are fulfilled if 3Pi/2 <= t <= 2 Pi. Look for example at

Plot[{x /. r -> 1, y /. r -> 1.5}, {t, 0, 2 Pi}]

The Volumeelement in Polar Coordinates (in R2 ) is dV = r dr dt, so your integral is written as

In[68]:= 1 / Pi Integrate[ r^2  r , {r, 0, 6}, {t, 3 Pi/2, 2 Pi}]

Out[68]= 162

Obviously your integral "lives" on a disk with radius r. Then you can equally write y as function of x and your integral is as well

In[77]:= 1/Pi Integrate[ x^2 + y^2, {x, 0, 6}, {y, 0, Sqrt[36 - x^2]}]

Out[77]= 162

Ok Frank, sorry! I attach the NMaximize.

Attachments:

DoubleIntegr.nb

Frank, What is the problem with the following?

NMaximize[{\!( *SubsuperscriptBox[([Integral]), (0), (6)]( *SubsuperscriptBox[([Integral]), (-6), (0)](( *SuperscriptBox[(x), (2)] + *SuperscriptBox[(y), (2)]))/[Pi]\ \ [DifferentialD]y [DifferentialD]x)),x^2+y^2<=36,y<=0},{x,y}]

The solution is: {275.02, {x -> 3.59228, y -> -0.0941189}}

It is rather unclear what you want to integrate. Below there are three alternatives with three different values: NMaximize[{[Integral]\!( *SubsuperscriptBox[([Integral]), ( *SuperscriptBox[(x), (2)] + *SuperscriptBox[(y), (2)]), (36)]( *FractionBox[(1), ([Pi])] (( *SuperscriptBox[(x), (2)] + *SuperscriptBox[(y), (2)])) [DifferentialD]x [DifferentialD]y\ )), x >= 0, y >= 0, x^2 + y^2 <= 36}, {x, y}]

Out[69]= {25789., {x -> 0., y -> 6.}}

NMaximize[{[Integral][Integral]1/[Pi] (x^2 + y^2) [DifferentialD]x [DifferentialD]y, x >= 0, y >= 0, x^2 + y^2 <= 36}, {x, y}]

Out[70]= {68.7549, {x -> 4.24264, y -> 4.24264}}

NMaximize[{\!( *SubsuperscriptBox[([Integral]), ( *SuperscriptBox[(x), (2)] + *SuperscriptBox[(y), (2)]), (36)]( *SubsuperscriptBox[([Integral]), ( *SuperscriptBox[(x), (2)] + *SuperscriptBox[(y), (2)]), (36)] *FractionBox[(1), ([Pi])] (( *SuperscriptBox[(x), (2)] + *SuperscriptBox[(y), (2)])) [DifferentialD]x [DifferentialD]y\ )), x >= 0, y >= 0, x^2 + y^2 <= 36}, {x, y}]

Out[72]= {-1.1913110^9, {x -> 2.0236310^-9, y -> 5.45121*10^-9}}