University Mathematics Handbook (2015)

VIII. Integral Calculus of Multivariable Functions

Chapter 3. Double Integral Calculation

a.  If the domain is rectangle , then:

b.  Let function be integrable in domain of the following properties:

1.   is bounded and closed.

2.  Any straight line parallel to -axis intersects the boundary of the region at no more than 2 points and , .

3.  Domain is bounded left and right between straight lines and , respectively.

Then

c.  If is integrable over bounded and closed region of the following properties:

1.   is bounded and closed. It is bounded above and below between the straight lines and respectively.

2.  Any straight line parallel to the -axis intersects the boundary of the region at no more than 2 points and , .

Then

d.  If region is more complex, it is divided into a finite number of domains of the shape of and .

Example: Illustrated area is divided into 3 regions by a straight line parallel to -axis.

All regions hold the condition specified in paragraph b, and therefore, the integral can be found using the formula, and: