NIntegrate and Integrate giving different results for ill-behaved function
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.
TL;DR Use HeavisideTheta's properties before integration.
This is my strategy.
First the HeavisideTheta gives you the following integration limits:
$$0\leq y \leq 1-x \qquad \& \qquad 0\leq x \leq 1$$
$$0\leq x \leq 1-y \qquad \& \qquad 0\leq y \leq 1$$
In both cases I used Integrate first then NIntegrate.
In the first case I could not integrate numerically beacuse of a divergence in the result of the symbolical integration; in fact I had:
Integrate[1/(100^2 x - y (1 - y) 90^2), {y, 0, 1 - x}]
Plot[%,{x,0,1}, PlotRange -> Full]
$$\dfrac{(\arctan(\frac{9}{\sqrt{-81 + 400 x}}) + \arctan({\frac{9 (1 - 2 x)}{\sqrt{-81 + 400 x}})}}{(\frac{450}{\sqrt{-81 + 400 x}})}$$
Maybe someone can come up with a solution also for this case.
In the second case I obtained for the symbolical integration:
Integrate[1/(100^2 x - y (1 - y) 90^2), {x, 0, 1 - y}]
Integrate[%, {y, 0, 1}]
N@%
(*0.0000600275 - 0.000314159 I*)
and for the numerical integration:
NIntegrate[
Integrate[1/(100^2 x - y (1 - y) 90^2), {x, 0, 1 - y}], {y, 0, 1}]
(*0.0000600275 - 0.000314159 I*)