## 3.5.9 Integrals over Regions

This does an integral over the unit circle.
 In[1]:=  Integrate[If[x^2 + y^2 < 1, 1, 0], {x, -1, 1}, {y, -1, 1}]
 Out[1]=
Here is an equivalent form.
 In[2]:=  Integrate[Boole[x^2 + y^2 < 1], {x, -1, 1}, {y, -1, 1}]
 Out[2]=

Even though an integral may be straightforward over a simple rectangular region, it can be significantly more complicated even over a circular region.

This gives a Bessel function.
 In[3]:=  Integrate[Exp[x] Boole[x^2 + y^2 < 1], {x, -1, 1}, {y, -1, 1}]
 Out[3]=

 Integrate[f Boole[cond], {x, , }, {y, , }] integrate f over the region where cond is True

Integrals over regions.

Particularly if there are parameters inside the conditions that define regions, the results for integrals over regions may break into several cases.

This gives a piecewise function of .
 In[4]:=  Integrate[Boole[a x < y], {x, 0, 1}, {y, 0, 1}]
 Out[4]=
With two parameters even this breaks into quite a few cases.
 In[5]:=  Integrate[Boole[a x < b], {x, 0, 1}]
 Out[5]=
This involves intersecting a circle with a square.
 In[6]:=  Integrate[Boole[x^2 + y^2 < a], {x, 0, 1}, {y, 0, 1}]
 Out[6]=
The region can have an infinite number of components.
 In[7]:=  Integrate[Boole[Sin[x] > 1/2] Exp[-x], {x, 0, Infinity}]
 Out[7]=
Integrate effectively does Lebesgue integration.
 In[8]:=  Integrate[Boole[Element[x, Rationals]], {x, 0, 1}]
 Out[8]=

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