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Homework Homework Assignment #11 Read Section 6.2 Page 379, Exercises: 1 – 53(EOO), 56 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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Homework
Homework Assignment #11 Read Section 6.2 Page 379, Exercises: 1 – 53(EOO), 56
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 3791. Find the area of the region between y = 4x + 4 and y = 3x 2 + 12 over [–3, 3]. (Figure 8)
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3 2
3
3 2
3
33 2
3
33 2
3
3 2 3 2
3 12 4 4
3 4 8
3 4 83 2
2 8
3 2 3 8 3 3 2 3 8 3
27 18 24 27 18 24 33 69 102
102
A x x dx
x x dx
x xx
x x x
A
Homework, Page 3795. Find the area between y = sin x and y = cos x over
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
0, .4
44
0 0
4
0
cos sin sin cos
2 2sin cos 0 1 2 1
2 2
2 1
A x x dx x x
x x
A
Homework, Page 379Let f (x) = 20 + x – x2 and g (x) = x2 – 5x .
9. Find the area of the region enclosed by the two graphs.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2 2 2 220 5 2 6 20 0 2 3 10 0
2 5 2 0 2,5
x x x x x x x x
x x x
Homework, Page 3799. Continued.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
5 52 2 2
2 2
52 3
2
3 32 2
20 5 20 6 2
20 6 22 3
5 2 20 5 3 5 2 20 2 3 2 2
3 3
250 16 266 100 75 40 12 175 28
3 3 3
609 266 343 343
3 3 3
A x x x x dx x x dx
x xx
A
Homework, Page 379
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2 2
1 113. Sketch the region between and for
1 11 1
and find its area.2 2
y yx x
x
1 12 2
1 12 2 22 2
11 2
12
1 12
1 1 1
2 2 2sin 2
6 6 3 3
dxA dx
x x x
x A
x
y
Homework, Page 379Find the area of the shaded region in the figure.
17.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2 2 2
30 22 23 02
1 1 1 31 1 1
2 2 4 2
1 11 1
2 2
0.208
x x x x x x
A x x x dx x x x dx
Homework, Page 379Find the area between the graphs of x = sin y and x = 1 – cos y over the given interval. (Figure 14)
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
21. 02
y
2 2
0 0
2
0
sin 1 cos sin cos 1
cos sin 0 1 1 0 02
2 22 2
A y y dy y y dy
y y y
A
Homework, Page 37925. Calculate the area enclosed by x = 9 – y2 and x = 5 in two ways: as an integral along the y-axis and as an integral along the x-axis.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2 2
232 22 2
2 22
3 3
93
29 9
5 5
5
5 9 4 2 9
9 5 4 43
2 2 8 8 32 4 2 4 2 8 8
3 3 3 3 3
99 9 2 9 2
32
4 32 32 0 8
3 3 3
y y y y x
yA y dy y dy x
xA x x dx xdx
A
x
y
Homework, Page 379Sketch the region enclosed by the curves and compute its area.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2 229. 4 , 4y x y x
2 2 2 2
2 22 2 2
2 2
23
3 3
2
4 4 2 8 4 2
4 4 8 2
2 2 8 2 8 2 2 8 2 2
3 3 3
16 16 96 32 64 64 16 16
3 3 3 3 3
x x x x x
A x x dx x dx
xx
A
Homework, Page 379Sketch the region enclosed by the curves and compute its area.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
333. 3 , 4 ,3
xy x y x y
3 4
3 3 4 4 3
33
1
43 9 3 ,4 4 3
3 3 3
3 4 4 3 4 3 0 1
4 3 4 23
2
A
A
x x xx x x A x x
x x x x x x x
xArea x x dx x dx
Area
x
y
Homework, Page 379Sketch the region enclosed by the curves and compute its area.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
237. , 6x y x y
2
2 22 2 2
2 2
6 0 2 3 0 3,2
446 6
3
44
3
y y y y y
A y y dy y y dy
A
x
y
Homework, Page 379Sketch the region enclosed by the curves and compute its area.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
241. 2 , 1 1x y x y
2 2
4 2
0
2 1 1 2 4 0 4 0 0,4
8 82 1 1
3 3
y y y y y y y y
A y y dy A
x
y
Homework, Page 379Sketch the region enclosed by the curves and compute its area.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2 345. sin , csc , ,
4 4y x y x x x
3 3
4 2 4
44
csc sin tan cos
2 2 1 1 2 2 2 2
2 2
A x x dx x x
A
x
y
Homework, Page 37949. Express the area (not signed) of the shaded region in Figure 16 as a sum of 3 integrals involving the functions f and g.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3 5 9
0 3 5A f x g x dx f x dx g x dx
Homework, Page 37956. The back of a guitar, Figure 17, has length 19 in. Measuring the width at 1-in intervals starting and ending ½-in from the end, obtain the values:
6, 9, 10.25, 10.75, 1075, 10.25, 9.75, 9.5, 10, 11.25, 12.75, 13.75, 14.25, 14.5, 14.5, 14, 13.25, 11.25, 9
Use midpoints to estimate the area of the back.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 37956. Continued
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
1 0 6 6 9 9 10.75 10.75 10.75 10.75 10.251
2 2 2 2 2 2
10.25 9.75 9.75 9.5 9.5 10 10 11.25 11.25 12.75 1
2 2 2 2 2
12.75 13.75 13.75 14.25 14.25 14.5 14.5 14.5 1
2 2 2 2
14.5 14 14 13. 1
2
A
2
25 13.25 11.25 11.25 9 1 9 0
2 2 2 2 2
211 in
