Integration over the Unit Disk

This computes the area of the unit disk using symbolic integration. Boole is a new function that converts inequalities to Boolean expressions.

(∫_(-∞)^∞)(∫_(-∞)^∞) Boole[
            (x^2) + (y^2) < 1] \[DifferentialD]y \[DifferentialD]x

This computes the same area using numeric methods, but using a similar symbolic reduction method as for the symbolic integral.

NIntegrate[
      Boole[(x^2) + (y^2) < 
          1], {x, -∞, ∞}, {y, -∞, ∞}]

Integration over Combinations of Inequalities

You can use any logical combination of inequalities and integrate a function over that region.

(∫_(-∞)^∞)(∫_(-∞)^∞)((y^4) + (x^2)) Boole[
              (x^2) + (y^2) < 1 && 
                x < y] \[DifferentialD]y \[DifferentialD]x

Integration over Parameterized Inequalities

With undefined parameters in the region description, the result is computed for all possible values.

(∫_(-1)^1)(∫_(-1)^1) Boole[
            a (x^2) + (y^2) < 
              1] \[DifferentialD]y \[DifferentialD]x

Note: //TraditionalForm automatically converts output to TraditionalForm.

(∫_(-1)^1)(∫_(-1)^1) Boole[
              a (x^2) + (y^2) < 1] \[DifferentialD]y \[DifferentialD]x // 
      TraditionalForm

Integration over an Infinite Collection of Intervals

The methods work generally for all regions described by polynomial inequalities, but also for many solvable regions described by transcendental inequalities.

In this case the integration is over an infinite collection of intervals.

(∫_(-∞)^∞) Boole[
              (1/2) < Sin[x] < (2/3)]/(x^2) \[DifferentialD]x