16-6 Calculus – Differentiation and Integration
2. ( ( , ) ( , )) ( , ) ( , ) .f x y g x y f x y dA g x y dA
RRR
± = ±
∫∫∫∫∫∫
3. f x y dA f x y R
R
( , ) ( , )
∫∫
≥ ≥0 0if on .
4. f x y dA g x y dA f x y g x y R
R R
( , ) ( , ) ( , ) ( , )
∫∫
≥
∫∫
≥if on .
5. f x y dA f x y dA f x y dA
R R R
( , ) ( , ) ( , ) .
∫∫
=
∫∫
+
∫∫
1 2
This holds when R is the union of two non-overlapping rectangles R
1
and R
2
.
Remark 16.1 The problems we are going to solve by integrating functions of two and three
variables are similar to the problems solved by single-variable integration. While solving the
problems we perform the necessary calculations by drawing our experience with functions of a
single variable. We shall see how