Transcript Slide 1

Homework
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Homework Assignment #13
Review Section 6.2
Page 389, Exercises: 25 – 33(EOO), 35
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 389
25.
Find the mass of a 2-m rod whose linear density function is
ρ ( x ) = 1 + 0.5 sin ( πx ) kg/m for 0 ≤ x ≤ 2.
2
M 
2
0
1


  x  dx   1  0.5sin  x   dx   x 
cos  x  
0
2

0
2
1
1
1
1

 

 2
cos 2    0 
cos 0   2 

2
2
2
2 2

 

M  2 kg
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 389
29.
Table 1 lists the population density (in people per square km) as a
function of distance r (in km) from the center of a rural town. Estimate the
total population within a 2-km radius of the center by taking the average
of the left- and right-endpoint approximations.
L10  0.2 125  102.3  83.8  68.6  56.2  46 
 0.2  37.6  30.8  25.2  20.7 
 119.24
L10  0.2 102.3  83.8  68.6  56.2  46 
 0.2  37.6  30.8  25.2  20.7  16.9 
 97.62
Avg . 
119.24  97.62
 108.43
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 389
33.
Find the flow rate through a tube of radius 4 cm, assuming
that the velocity of fluid particles at a distance r cm from the center
is v (r) = 16 – r2 cm/s.
Q  2 
4
0
4
 r
r 
r 16  r  dr  2  16r  r  dr  2 16  
0
 2 4 0
2
4
2
4
3
4

 
4


2
 2   8  4  
  0   2 128  64   128
 

4
 

Q  128 cm3 /s
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 389
35.
A solid rod of radius 1 cm is placed in a pipe of radius 3
cm so their axes are aligned. Water flows through the pipe and
around the rod. Find the flow rate if the velocity is given by the
radial function v (r) = 0.5(r – 1)(3 – r) cm/s.
Q  2  r  v  r   dr  2  r  r  1 3  r  dr
3
1
3
1
 2   r 3  4r 2  3r  dr
3
1
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 389
35.
Continued.
3
 r
r
r 
Q  2    r  4r  3r  dr  2    4  3 
1
3
2 1
 4
3
4
3
3
2
2
3
2
4
3
2
   34



3
3
1
1
1










 2   
4
3
4
3


  4

 4
3
2
3
2
 


  81
27   1 4 3  
4

 2     36           2  20  36  12  
2   4 3 2 
3

 4
16
Q
cm3 /s
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Average Values
If we drive for five hours and cover 300 miles, we would say our
average speed was 60 mph. Graphically, it might look like this:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Remembering that the integral gives us the area between the graph
of the function and the x-axis on the interval [a, b], dividing the area
by the width (b – a) will give us the average value of f (x) on [a, b
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
As illustrated in Figure 12, the area under the graph of f (x) = sin x on
[0, π] is the same as the are of the rectangle with length π and width
2/π.


0
sin xdx   cos x 0    cos   cos 0     1  1  2

Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 389
Calculate the average over the given interval.
40. f  x   sec2 x, 0,  
4

Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 389
Calculate the average over the given interval.
46. f  x   e nx ,  1,1
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 389
58.
Let M be the average value of f (x) = x4 on [0, 3]. Find a
value of c in [0, 3] such that f (c) = M.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework
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Homework Assignment #14
Read Section 6.3
Page 389, Exercises: 37 – 59(Odd)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company