Transcript Slide 1

Chapter 15 – Multiple Integrals
15.3 Double Integrals over General Regions
Objectives:
 Use double integrals to find
the areas of regions of
different shapes
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Double Integrals

For double integrals, we want to be able to
integrate a function f not just over
rectangles but also over regions D of more
general shape.
◦ One such shape
is illustrated.
15.3 Double Integrals over General
Regions
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Double Integrals

We suppose that D is a bounded
region.
◦ This means that D can be enclosed in a rectangular
region R as shown.
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Regions
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Double Integrals

Then, we define a new function F
with domain R by:
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Regions
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Double Integral

If F is integrable over R, then we define the
double integral of f over D by:
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Regions
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Double Integrals

You can see that this is reasonable by:
◦ Comparing the graphs of f and F here.
◦ Remembering F ( x, y) dA is the volume under
the graph of F. R

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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)}
where g1 and g2 are continuous on [a, b].
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Type I Examples
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Type I Regions
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Type II Regions

We also consider plane regions of type II, which
can be expressed as:
where h1 and h2 are continuous.
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Type II Regions

So, similar to Equation 3, we have
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Example 1

Evaluate the double integral.
2
2
x
y

x
dA, D   x, y  | 0  y  1, 0  x  y

D
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Example 2 – pg. 995 #22

Evaluate the double integral.
2
xydA

D
D is the triangular region with
vertices (0,0), (1,2), and (0,3).
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Example 3 – pg. 996 # 26

Find the volume of the given solid.
Enclosed by the paraboloid z  x  3 y
and the planes x=0, y=1, z=0, and
y=x.
2
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Example 4 – pg. 996 # 28

Find the volume of the given solid.
Bounded by the planes z=x, y=x, z=0,
and x + y = 2.
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Regions
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Example 5 – pg. 996 # 32

Find the volume of the given solid.
Bounded by the cylinders
x  y  r and y  z  r
2
2
2
2
2
2
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Example 6 – pg. 996 # 50

Evaluate the integral by reversing the
order of integration.


  cos  x dx dy
2
0
y
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Example 7 – pg. 996 # 52

Evaluate the integral by reversing the
order of integration.
1 1
e
x/ y
dy dx
0 x
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Example 8 – pg. 996 # 54

Evaluate the integral by reversing the
order of integration.
8 2

0
3
x4
e dx dy
y
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More Examples
The video examples below are from
section 14.6 in your textbook. Please
watch them on your own time for
extra instruction. Each video is
about 2 minutes in length.
◦ Example 1
◦ Example 3
◦ Example 5
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Demonstrations

Feel free to explore these
demonstrations below.
◦ Double Integral for Volume
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