3.5.9 Integrals over Regions| This does an integral over the unit circle. | |
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Integrate[If[x^2 + y^2 < 1, 1, 0], {x, -1, 1}, {y, -1, 1}]
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| Here is an equivalent form. | |
In[2]:=
Integrate[Boole[x^2 + y^2 < 1], {x, -1, 1}, {y, -1, 1}]
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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. | |
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Integrate[Exp[x] Boole[x^2 + y^2 < 1],
{x, -1, 1}, {y, -1, 1}]
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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  . | |
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Integrate[Boole[a x < y], {x, 0, 1}, {y, 0, 1}]
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| With two parameters even this breaks into quite a few cases. | |
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Integrate[Boole[a x < b], {x, 0, 1}]
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| This involves intersecting a circle with a square. | |
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Integrate[Boole[x^2 + y^2 < a], {x, 0, 1}, {y, 0, 1}]
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| The region can have an infinite number of components. | |
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Integrate[Boole[Sin[x] > 1/2] Exp[-x],
{x, 0, Infinity}]
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| Integrate effectively does Lebesgue integration. | |
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Integrate[Boole[Element[x, Rationals]], {x, 0, 1}]
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