Double%20Integrals%20over%20General%20Regions

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Section 16.3

• Double Integrals over General Regions

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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
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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
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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
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PROPERTIES OF DOUBLE INTEGRALS
3. If f (x, y) g(x, y) for all (x, y) in D,
then
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PROPERTIES (CONTINUED)
4. If D D1 U D2, 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).
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PROPERTIES (CONCLUDED)
6. If m f (x, y) M for all (x, y) in D,
then