Section 16.3

1. Section 16.3 Double Integrals over General Regions

2. DOUBLE INTEGRAL OVER A GENERAL REGION Suppose that D is a bounded region in the plane; that is, D can be enclosed by a rectangular region R. We define a new function F with domain R by If the double integral of F exists over R, then we define the double integral of f over D by

3. TYPE I REGIONS A plane region D is said to be of type I if it lies between the graphs of two continuous functions of x. That is, D = {(x, y) | a≤ x ≤ b, g1(x) ≤ y ≤ g2(x) } If f is continuous on a type I region D as described above, then

4. TYPE II REGIONS A plane region D is said to be of type II if it lies between the graphs of two continuous functions of y. That is, D = {(x, y) | c≤ y ≤ d, h1(y) ≤ x ≤ h2(y) } If f is continuous on a type II region D as described above, then

5. PROPERTIES OF DOUBLE INTEGRALS 3. If f (x, y) ≥ g(x, y) for all (x, y) in D, then

6. PROPERTIES (CONTINUED) 4. If D = D1UD2, where D1 and D2 do not overlap except perhaps on their boundaries, then 5. If we integrate the constant function f (x, y) = 1 over a region D, we get the area of D [A(D)].

7. PROPERTIES (CONCLUDED) 6. If m≤ f (x, y) ≤ M for all (x, y) in D, then