Domain of integration ($D$)

The domain of integration is the set $D$ over which an integral is taken. For a multiple integral, inequalities or parameter ranges specify which points belong to $D$; a change of variables transforms both these bounds and the integration element through its Jacobian determinant. For example, a polar region $c\le r\le 1+\cos \varphi$ exists only at angles where $1+\cos \varphi \ge c$.

The set over which an integral is taken. For a double integral, boundary inequalities describe the allowed coordinate pairs, and slices of this set determine the limits in an iterated integral. The domain need not be a rectangle or connected. A change of variables transforms both the integrand and this set.