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Homework Assignment #13 Review Section 6.2 Page 389, Exercises: 25 – 33(EOO), 35
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 38925. 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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2
2 2
0 00
11 0.5sin cos
2
1 1 1 12 cos 2 0 cos0 2 2
2 2 2 2
2 kg
M x dx x dx x x
M
Homework, Page 38929. 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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
10
10
0.2 125 102.3 83.8 68.6 56.2 46
0.2 37.6 30.8 25.2 20.7
119.24
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
119.24 97.62. 108.43
2
L
L
Avg
Homework, Page 38933. 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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
42 4
4 42 3
0 00
42
3
2 16 2 16 2 162 4
42 8 4 0 2 128 64 128
4
128 cm /s
r rQ r r dr r r dr
Q
Homework, Page 38935. 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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3 3
1 1
3 3 2
1
2 2 1 3
2 4 3
Q r v r dr r r r dr
r r r dr
Homework, Page 38935. Continued.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
34 3 2
3 3 2
11
4 3 2 4 3 2
3
2 4 3 2 4 34 3 2
3 3 3 1 1 12 4 3 4 3
4 3 2 4 3 2
81 27 1 4 3 42 36 2 20 36 12
4 2 4 3 2 3
16 cm /s
3
r r rQ r r r dr
Q
Average ValuesIf 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 CalculusCopyright © 2008 W. H. Freeman and Company
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Remembering that the integral gives us the area between the graphof the function and the x-axis on the interval [a, b], dividing the areaby the width (b – a) will give us the average value of f (x) on [a, b
Rogawski CalculusCopyright © 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.
00sin cos cos cos 0 1 1 2xdx x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Example, Page 389Calculate the average over the given interval.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
240. sec , 0, 4f x x
Example, Page 389Calculate the average over the given interval.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
46. , 1,1nxf x e
Example, Page 38958. 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 CalculusCopyright © 2008 W. H. Freeman and Company
Homework
Homework Assignment #14 Read Section 6.3 Page 389, Exercises: 37 – 59(Odd)
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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