高等数学英文版课件PPT 13 Multiple Integrals.ppt

Multiple Integrals
Chapter 13
by Zhian Liang
Double integrals over rectangles
Suppose f(x) is defined
The properties of the double integrals
then
EXERCISES
Page 837
1. 3.
Iterated Integrals
to calculate
The double integral can be obtained by evaluating two single integrals.
The steps to calculate
, where
Then calculate
with respect to
Fix
(1)
(called iterated integral)
(2)
(3)
Similarly
EXAMPLE 1 Evaluate the iterated integrals
(See the blackboard)
(4) Fubini’s Theorem If is continuous on the rectangle then
More generally, this is true if we assume that
is bounded on , is discontinuous only on
a finite number of smooth curves, and the iterated
integrals exist.
Interpret the double integral as the volume V of the solid
where
is the area of a cross-section of S in the plane through x perpendicular to the x-axis.
Similarly
EXAMPLE 2
Evaluate the double integral
where
(See the blackboard)
EXAMPLE 3
Evaluate
,where
Solution 1 If we first integrate with respect to x,
we get
Solution 2 If we first integrate with respect to y, then
EXAMPLE 4 Find the volume of the solid S that is bounded by the elliptic paraboloid , the plane and , and three coordinate planes.
We first observe that S is the solid that lies under the surface
and the above the
Square
Therefore,
Solution
If on , then
EXAMPLE 5
, then
If
EXERCISES
Page 842
1(2), 6, 10, 16, 17,
Double integrals over general regions
To integrate over general regions like
which is bounded, being enclosed in a rectangular region R .
Then we define a new function F with domain R by
(1)
if
if
If F is integrable over R , then we say f is integrable over D and we define the