Chabot Mathematics
§7.6 Double
Integrals
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Review §
7.5
 Any QUESTIONS About
• §7.5 → Lagrange Multipliers
 Any
QUESTIONS
About
HomeWork
• §7.5 → HW-8
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Partial-Deriv↔Partial-Integ
 Extend the Concept of a “Partial”
Operation to Integration.
2

z
 Consider the
 h  x, y 
nd
mixed 2 Partial xy
 ReWrite the Partial in Lebniz Notation:
2 z
 z

z y   hx, y 


xy
x y
x
 Now Let: z y  ux, y 
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Partial-Deriv↔Partial-Integ
 Thus with ∂z/∂y = u:


u
z y  
u 
 h  x, y 
x
x
x
 Now Multiply both sides by ∂x and
Integrate
 u 
 x  x  h x, y x   1 u   h x, y   x
 Integration with respect to the Partial
Differential, ∂x, implies that y is held
CONSTANT during the AntiDerivation
Chabot College Mathematics
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Partial-Deriv↔Partial-Integ
 Performing The AntiDerivation while not
including the Constant find:
z
1 u  u  y   hx, y  x
 Now Let: g x, y    hx, y   x
 Then substitute, then multiply by ∂x
 z

z
 g  x, y 

 g x, y   y

y
 y

Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Partial-Deriv↔Partial-Integ
 Integrating find:
 1 z

 After AntiDerivation: z 
 But ReCall: g x, y  
 Back Substituting find
z   g x, y  y 
 g x, y  y
 g x, y  y
 hx, y  x
  hx, y  x y
 By the Associative Property


2
 from
z    h x, y   x  y
 h x, y 
xy


Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Partial-Deriv↔Partial-Integ
2

z
 Also ReCall
Clairaut’s Theorem: xy
2 z

yx
 This Order-Independence also Applies
to Partial Integrals Which leads to the
Final Statement of the Double Integral
z  x, y   C

  hx, y  x  y

  hx, y  y  x
• C is the Constant of Integration
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Area BETWEEN Curves
 As before Find Area
by adding Vertical
Strips.
 In this case for the
Strip Shown:
• Width = Δx
• Height = ytop − ybot or
Hgt  f xk   g xk 
ytop  f  xk 
Astrip
x
ybot  g xk 
xk
 Then the strip area
Hgt  Wdh   f xk   g xk  x
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Area BETWEEN Curves
 Note that for every
CONSTANT xk, that
y runs: g xk   f xk 
y
 Now divide the Hgt
into pieces Δy high
A
 So then ΔA:
A  y  x
x
 Then Astrip is simply
the sum of all the
y
y

small boxes
Astrip   y  x    y   x
Chabot College Mathematics
9
to p
to p
yb o t
 yb o t

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Area BETWEEN Curves
 Substitute: ybot  g xk 
ytop  f  xk 
 Then
 f  xk  
Astrip    y   x
 g  xk  
 Next Add Up all the
Strips to find the
Total Area, A
x b
xa
f x 


 k 

A   Astrip     y   x 

x a 
 g  xk  

x b
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Area BETWEEN Curves
 This Relation
A   A
x b  f  xk 



A     y   x 

xa 
 g  xk  
x  b f  xk 
A
y  x

 
x  a g xk
 Is simply a Riemann
Sum
 Then in the Limit
y  0
x  0
Chabot College Mathematics
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 Find
A 
x b y  f  xk 
 
x  a y  g  xk 
 dA
A 
A 
1 dy  dx
R
 1 dy  dx
R
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Area Between Curves
 Find the area of the region contained
between the parabolas
MTH16 • Bruce Mayer, PE
y 9 x
2
y  x 1
2
10
2
-x + 9
9
2
y = f(x) = -x2 + 9 & x2 + 1
x +1
8
7
6
5
Over
2 x  2
4
3
2
1
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x
Chabot College Mathematics
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
MTH16 • Bruce Mayer, PE
10
-x2 + 9
y = f(x) = -x2 + 9 & x2 + 1
9
x2 + 1
8
7
6
5
4
3
2
1
0
-2
Chabot College Mathematics
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-1.5
-1
-0.5
0
x
0.5
1
1.5
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Area Between Curves
 SOLUTION: Use Double Integration

 1 dy dx 
 2 1 x 2
2
MTH16 • Bruce Mayer, PE
 y y 1 x
y 9  x 2
2
10
dx
2
2
x2 + 1
8
7
 8  2 x  dx
 8 x  x 
 8(2)  (2)  8(2) 

2
2
2
3 2
2
3
2
Chabot College Mathematics
14
-x2 + 9
9
y = f(x) = -x2 + 9 & x2 + 1
2 9 x 2
5
4
A  21.33 unit
3
2
2
1
0
-2
3
3
6
-1.5
-1
-0.5
0
0.5
x
2
1

1.5
2
3
64
(

2
)

3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
3
Chabot College Mathematics
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
MATLAB code
% Bruce Mayer, PE
% MTH-16 • 22Feb14
% Area_Between_fcn_Graph_BlueGreen_BkGnd_Template_1306.m
% Ref: E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E.
% Herhold, G. C. Gregory, "An Engineer's Guide to MATLAB", ISBN
% 978-0-13-199110-1, Pearson Higher Ed, 2011, pp294-295
%
clc; clear; clf; % Clear Figure Window
%
% The Function
xmin = -2; xmax = 2;
ymin = 0; ymax = 10;
x = linspace(xmin,xmax,500);
y1 = -x.^2 + 9;
y2 = x.^2 + 1;
%
plot(x,y1,'--', x,y2,'m','LineWidth', 5), axis([0 xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = -x^2 + 9 &
x^2 + 1'),...
title(['\fontsize{16}MTH16 • Bruce Mayer, PE',]),...
legend('-x^2 + 9','x^2 + 1') %
display('Showing 2Fcn Plot; hit ANY KEY to Continue')
% "hold" = Retain current graph when adding new graphs
hold on
disp('Hit ANY KEY to show Fill')
pause
%
xn = linspace(xmin, xmax, 500);
fill([xn,fliplr(xn)],[-xn.^2 + 9, fliplr(x.^2 + 1)],[.49 1 .63]), grid on
% alternate RGB triple: [.78 .4 .01]
Volume Under a Surface
 Use Long Strips to
find the Area under
a Curve (AuC) by
Riemann
Summation
 Use Long Boxes
to find
the
Volume
Under a
Surface
(VUS) by Riemann
Summation
Vol  lim
 f x , y y  x
M
N
x 0
j 1 k 1
y 0
xm ym
Vol  
0
Chabot College Mathematics
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
0
j
j
f x, y  dy  dx
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
VUS by Double Integral
V  z  A
V   f  x, y  y  x 
Vtot   V
Vtot   z  A
Vtot    f x, y  y  x 
x
A  y  x
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
y
Example  Vol under Surf
 Find the volume
under the Surface
described by
z  xe
 Over the Domain
x :  1  1
y : 1 4
z = f(x,y)=x2e-y
2 y
MTH16 • Bruce Mayer, PE
0.3
0.2
0.1
0
4
1
3
0.5
0
2
 See Plot at Right
Chabot College Mathematics
18
y
MTH15 3Var 3D Plot.m
-0.5
1
-1
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Vol under Surf
 SOLUTION: Find Vol by Double Integral
4 1
V    z  dx dy
1 1
1


2 y
2 y
1 1 x e dx dy  1 1 x e dx dy
4 1
4
x 1
1 3  y 
   x e  dy
3
 x  1
1 
4
 1  y  1  y 
   e    e dy
3
 3

1 
4
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Vol under Surf
 Completing the Reduction
 1  y  1  y 
2 y 
1  3 e    3 e dy  1  3 e dy
4
4
 
z = f(x,y)=x2e-y
MTH16 • Bruce Mayer, PE
2 y
 e
3
VuS  0.233
0.3

y 1
2  4 1
  e e
3
0.2
0.1
0
4
1
3

 0.233
0.5
0
2
y
MTH15 3Var 3D Plot.m
-0.5
1
Chabot College Mathematics
20
y 4
-1
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
VUS for NonRectangular Region
 If the Base Region, R, for a Volume
Integral is NonRectangular and can be
described by InEqualities
a  x  b and
g1 x  y  g2 x
 Then by adding up all the long boxes
x b
yg x 

  dx




f
x
,
y

dA

V

f
x
,
y

dy
R
xa y g  x 

2
 If R described by


1
h1  y   x  h2  y  and c  y  d
 Then: R f x, y  dA
Chabot College Mathematics
21
V 

x d
y c
 x h2  y  f x, y   dx   dy
 x  g1  y 

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  NonRectangular VUS
 Find the volume under the surface
x2 y
z  f x, y   x  e
 Over the Region Bounded by
x0
y0
x y 5
 SOLUTION: First,
visualize the limits
of integration using
a graph of the
Base Plane
Integration Region:
Chabot College Mathematics
22
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  NonRectangular VUS
 The outer limits of integration need to
be numerical (no variables), but the
Inner limits can contain expressions in
x (or y) as in the definition.
 In this case, choose the inner limits to
be with respect to y, then find the limits
of the y values in terms of x
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  NonRectangular VUS
 Each y-value in the region is restricted
by the constant height 0 at the top, at
the bottom by the Line:
x  y  5  y  x 5
V
 Thus the Double Integral (so far):
x b
yg x
x b
y 0




  
f x, y   dy  dx   
x  e x  2 y  dy   dx


x a  y  g  x 
x  a  y  x 5
2

1



 In Simplified V  x  e x 2 y  dy  dx


Notation
a x 5
b 0
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  NonRectangular VUS
 Now, Because the outer integral needs
to contain only numbers values,
consider only the absolute limits on
the x-values in the figure:
• a MINimum of 0 and a MAXimum of 5
 Thus the Completed Double Integral
 x  e  dy  dx
5 0
V 
x2 y
0 x 5
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  NonRectangular VUS
 Complete the Mathematical Reduction
5 0
 xe
0 x 5
x2 y
5

dy dx   xy 
0
5

1
2
e

x  2 y y 0
y  x 5
dx
 e  x( x  5) 
1
x
2
1

x  2 ( x 5 )
e
dx
2
0
5
   x 2  5x  1 2 e x  1 2 e3 x 10 dx
0

V 
Chabot College Mathematics
26
1
x 
3
3
5
x 
2
2
1
e 
x
2
1
6
e

3 x 10 5
0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  NonRectangular VUS
 Complete the Mathematical Reduction

 
V 
1
1
x 
3
3
5
x 
2
2
1
e 
x
2
1
6
e

3 x 10 5
0

3
5 (5) 2  1 e 5  1 e3( 5) 10 
(
5
)

3
2
2
6

1
3
5 (0) 2  1 e 0  1 e3( 0 ) 10
(
0
)

3
2
2
6
V  69.80
 The volume contained underneath the
surface and over the triangular region in
the XY plane is approximately 69.8
cubic units.
Chabot College Mathematics
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx

Example  NonRectangular VUS
 Verify Constrained VUS by MuPad
V
:= int((int(x+E^(x+2*y), y=x-5..0)), x=0..5)
Vnum = float(V)
Chabot College Mathematics
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Average Value
 Recall from Section
5.4 that the average
value of a function f
of one variable
defined on
an interval [a, b] is
1 b
V
f  x  dx

ba a
Chabot College Mathematics
29
 Similarly, the
average value
of a function f of two
variables defined on
a rectangle R to be:
1
AV 
f  x, y  dA

area of R R
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Average Sales
 Weekly sales of a new product depend
on its price p in dollars per item and
time t in weeks after its release, can be
2
Modeled by: S(p, t) =160 - pt - 0.3p
• Where S is measured in k-units sold
 Find the average weekly sales of the
product during the first six weeks after
release and when the product’s price
varies between 15 – t and 25 – t.
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Average Sales
 SOLUTION: first find the area of the
region of integration as shown below
 Note that The price
Constraints produce
a Parallelogram-like
Region
 By the Parallelogram
Area Formula
A  Bh  6 10  60
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Average Sales
 Proceed with the Double Integration
6 25t
1
AV    S  p, t   dp  dt
A 0 15t
6 25t


1
   160  pt  0.3 p 2  dp  dt
60 0 15t
6

1
2
3
160
p

0
.
5
p
t

0
.
1
p
60 0
Chabot College Mathematics
32

p  25t
p 15t
dt
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Average Sales
 Continue the Double Integration
6

1
2
3

160
(
25

t
)

0
.
5
(
25

t
)
t

0
.
1
(
25

t
)
60 0


 160(15  t )  0.5(15  t ) 2 t  0.1(15  t ) 3  dt
6


1
  7t 2  80t  375  dt
60 0
6


1
1
2
7t  80t  375  dt 

60 0
60
Chabot College Mathematics
33

7
3

6
t  40t  375t 0
3
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx

Example  Average Sales
 Complete the Double Integration
1

60

7

1
3 (6)  40(6)  375(6) 
60
3
2

7
3
2
(
0
)

40
(
0
)
 375(0)
3
 21.9
 The average weekly sales is 21,900
units over the time and pricing
constraints given.
Chabot College Mathematics
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx

WhiteBoard Work
 Problems From §7.6
• P7.6-89 → Exposure
to Disease
Chabot College Mathematics
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
All Done for Today
Volume by
Riemann
Sum
Chabot College Mathematics
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
2a
–
Chabot College Mathematics
37
BMayer@ChabotCollege.edu
2b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
§7.3 Learning Goals
 Define and compute double integrals over
rectangular and NONrectangular regions in
the xy plane
 Use double integrals
in problems involving
• Area
• Volume,
• Average Value
• Population Density
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38
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Chabot College Mathematics
39
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Chabot College Mathematics
40
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Chabot College Mathematics
41
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Chabot College Mathematics
42
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx