3.4.4. Change the order of integration

Let a double integral be given . If the integration domain is D (Fig. 15), given by the inequalities is also correct with respect to the OS axis, i.e. the boundary of the domain D is intersected by the straight line y = c ( c constant) at no more than two points, then the domain D can be given by other inequalities:

.

Here α and β are the largest and smallest values ​​of y in the domain D , respectively;
x = ₁ ( y ) is the left side of the boundary;
x = ψ ₂ ( y ) is the right side of the boundary of the domain D.

Then in the double integral you can change the order of integration:

Fig. 15