Question with Integrate and Region

Question

Let's say I've got a 2D region r=Region[Polygon[{{0,0},{1,0},{1,1}}]]. I can do integrals like Integrate[1,{x,y}\[Element]r], or Integrate[y,{x,y}\[Element]r], to calculate areas and area moments.

I'd like to do an integral like Integrate[c^2,{x}\[Element]r], where c is the height (ymax-ymin) of the region at a particular value of x.

Any ideas appreciated.

-David

Answer 1

You can define a height of a region as its intersection with a vertical line at $x$:

c[ℛ_Region][x_?NumericQ] := 
 ArcLength[RegionIntersection[ℛ, InfiniteLine[{x, 0}, {0, 1}]]]

Then compute its bounds and integrate:

ℛ = Region@Annulus[]
{{xmin, xmax}, {ymin, ymax}} = RegionBounds[ℛ];
NIntegrate[c[ℛ][x]^2, {x, xmin, xmax}]