Integrate over implicit 1D region: works for algebraic but not transcendental equation?

Question

QUESTION

How come this works:

NIntegrate[1, {x, y} ∈ 
  ImplicitRegion[{x == y^3, x <= 1, x >= 0}, {x, y}]]

But this fails:

NIntegrate[1, {x, y} ∈ 
  ImplicitRegion[{Sin[Pi x] == y, x <= 1, x >= 0}, {x, y}]]

where we have

ContourPlot[x == y^3, {x, 0, 1}, {y, 0, 1}]

and

ContourPlot[Sin[Pi x] == y, {x, 0, 1}, {y, 0, 1}]

A related question is this one

Answer 1

reg = ImplicitRegion[{Sin[Pi x] == y, x <= 1, x >= 0}, {x, y}]

This is a 1D region embedded in 2D space. NIntegrate needs to know this to produce a reasonable result. My guess is that it uses RegionDimension, which fails here:

In[41]:= RegionDimension[reg]

During evaluation of In[41]:= RegionDimension::nmet: Unable to compute the dimension of region ImplicitRegion[Sin[π x]==y&&x<=1&&x>=0,{x,y}]. >>

Out[41]= RegionDimension[
 ImplicitRegion[Sin[π x] == y && x <= 1 && x >= 0, {x, y}]]

RegionMeasure would be equivalent to your NIntegrate example in this specific case and it also fails.

Mathematica can determine the dimension of the other region just fine:

RegionDimension@
 ImplicitRegion[{x == y^3, x <= 1, x >= 0}, {x, y}]
(* 1 *)

We could however discretize the region first:

In[42]:= dreg = DiscretizeRegion[reg]

In[43]:= RegionDimension[dreg]
Out[43]= 1

In[44]:= RegionMeasure[dreg]
Out[44]= 2.30414

In[45]:= NIntegrate[1, {x, y} ∈ dreg]
Out[45]= 2.30414

This works, but the problem is that at the discretization step we lose precision that NIntegrate can never re-gain afterwards.

DiscretizeRegion has several options which can control how and how accurately the discretization is done. I'm not sure which one is the best choice in this case, but MaxCellMeasure would be one.