calculus and analysis - NIntegrate and Integrate giving different results for ill-behaved function

I'm trying to integrate the following function with Mathematica 8:

$$ I(a,b)= \int_0^1 \mathrm{d}x\int_0^1\mathrm{d}y \,\theta(1-x-y) \frac{1}{x a^2-y(1-y)b^2},$$ where $\theta$ is the Heaviside function. However, I find different results with Integrate or NIntegrate and I don't understand why.

More specifically, for a=100 and b=90:

NIntegrate[HeavisideTheta[1 - x - y]/(x a^2 - y (1 - y) b^2), {x, 0, 1}, {y, 0, 1}]

gives

0.0000927294,

while

Integrate[HeavisideTheta[1 - x - y]/( x a^2 - y (1 - y) b^2), {x, 0, 1}, {y, 0, 1}, PrincipalValue -> True]

gives

+0.0000600275+0.000314159 I.

What is the correct result? Why does Integrate give a complex result?

Answer

Reversing the order of integration produces a solution:

ans= Integrate[HeavisideTheta[1 - x - y]/(x 100^2 - y (1 - y) 90^2), {y, 0, 1}, {x, 0, 1}, 
  PrincipalValue -> True]
(* Log[(100*10^(38/81))/(81*19^(19/81))]/10000 *)

N[ans]
(* 0.000060027526501455836 *)

Solutions of this sort are what I would expect based on outlining a pencil-and-paper derivation.

Attempting to solve the integral numerically produces error messages. For instance,

NIntegrate[HeavisideTheta[1 - x - y]/(x 100^2 - y (1 - y) 90^2), {y, 0, 1}, {x, 0, 1}, 
  MinRecursion -> 30, MaxRecursion -> 60]

produces

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>
NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 0.0000591404833446268` and 0.0003356281358114434` for the integral and error estimates. >>
(* 0.0000591404833446268 *)

Of course,NIntegrate has many options, and one or more of them may produce an acceptable answer.

Comments

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