3.4.3. Calculation of the double integral

If the boundary of the region D is intersected by any straight line x = c ( c - const ) at no more than two points, then the domain D is called regular in the direction of the axis OX . The bounded domain D , correct in the direction of the OX axis, is given by the inequalities:

a ≤ x ≤ b , φ 1 ( x ) ≤ y ≤ φ 2 ( x ) ,

where φ 1 ( x ) , φ 2 ( x ) are functions that are continuous on the interval [ a, b ] .

In this case, the double integral reduces to a double integral .
First, assuming that the integration variable x is constant, we find a definite integral as a function of f ( x ) variable x .
Then we find a definite integral .