Transcript Slide 1
Homework
Homework Assignment #47
Read Section 7.1
Page 398, Exercises: 23 – 51(Odd)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 398
Find the volume of the solid obtained by rotating region A in
Figure 10 about the given axis.
23. x-axis
V
2
0
6 x 2 dx
2
2
2
36 x 4 4 x 2 4 dx
2
0
2
x
x
32 32
32 x 4 64 0
5
3 0
5
3
5
3
704
960 96 160 704
V
15
15
15
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 398
Find the volume of the solid obtained by rotating region A in
Figure 10 about the given axis.
25. y 2
V
2
0
2 6 2 x 2 dx
2
2
2
16 x 4 dx
2
0
2
x
32
16 x 32 0
5 0
5
5
128
160 32 128
V
5
5
5
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 398
Find the volume of the solid obtained by rotating region A in
Figure 10 about the given axis.
27. x 3
y x2 2 y 2 x2 x
y23
V
6
2
6
2
y 2 3 0 3 dy
2
2
y 2 9 9 dy
y2
y 2
2y 3
3
2
2
y2
3
6
2
2
18 12 2 8 2 4 0 24 V 24
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 398
Find the volume of the solid obtained by rotating region B in
Figure 10 about the given axis.
29. x-axis
V x 2 dx x 4 4 x 2 4 dx
2
2
2
2
0
0
2
x
x
4 4x
3
5
0
5
3
32 32
8 0
3
5
376
96 160 120 376
V
15
15
15
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 398
Find the volume of the solid obtained by rotating region B in
Figure 10 about the given axis.
31. y 6
V 6 0 6 x 2 dx 36 x 4 8 x 2 16 dx
2
2
0
2
2
2
0
2
x
x
32 64
20 x 8 40 0
5
3 0
5
3
5
3
824
600 96 320 824
V
15
15
15
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 398
Find the volume of the solid obtained by rotating region B in
Figure 10 about the given axis.
33. x 2
y x2 2 x2 y 2 x
V 2 2 2 y 2
6
2
2
8
6
2
y2
2
dy
y 2
4 4 y 2 y 2 dy 8 2 y 4
3
2
3
6
2
y2
2
2
8
8
8 12 8 18 4 0 2
3
3
32
72 64 32
8
V
3
3
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 398
Find the volume of the solid obtained by rotating the region
enclosed by the graphs about the given axis.
35. y x 2 , y 12 x, x 0, about y 2
x 2 12 x x 2 x 12 0 x 4 x 3 0 x 4,3
V
3
0
3
0
12 x 2 x 2 dx
2
2
196 28x x x
2
4
2
4 x 2 4 dx
3
x
2
3
192 28 x 3 x x dx 192 x 14 x x
0
5 0
3
5
2
4
243
2880 630 135 243
576 126 27
5
5
1872
1872
V
5
5
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 398
Find the volume of the solid obtained by rotating the region
enclosed by the graphs about the given axis.
37. y 16 x, y 3x 12, x 0, about y-axis
16 x 3x 12 4 x 4 x 1 y 15
2
16
2
y
V 4 dy 16 y dy
12
15
3
15
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
37.
Homework, Page 398
Continued.
2
16
2
y
V 4 dy 16 y dy
12
15
3
15
16
y2 8
y 16 dy 256 32 y y 2 dy
12
15
9 3
15
15
16
y 4 2
y
2
y 16 y 256 y 16 y
3 15
27 3
12
3
3
3375
1728
300 240
192 192
27
27
4096
3375
4096 4096
3840 3600
3
3
3375 1620 1728 4096 720 3375
27 9 4
27
3
27 3
Homework, Page 398
Find the volume of the solid obtained by rotating the region
enclosed by the graphs about the given axis.
9
39. y 2 , y 10 x 2 , about x-axis
x
9
2
4
2
10
x
x
10
x
9 0 x 1, 3
2
x
V 10 x
3
1
2 2
2
3
9
2 dx 100 20 x 2 x 4 81x 4 dx
1
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 398
39. Continued.
V 10 x
3
1
2 2
2
3
9
2 dx 100 20 x 2 x 4 81x 4 dx
1
x
3
3
x
x
x
20 x
x 27
100 x 20 81
3
100 x
3 5
3 1
3
5 x 1
3
5
3
3
5
243
20 1
300 180
1 100 27
5
3 5
2544 1808 736
600 243 5 1500 100 3 405
5
15
15
15
Two regions enclosed by the curves, means two volumes V
1472
15
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 398
Find the volume of the solid obtained by rotating the region
enclosed by the graphs about the given axis.
1
5
41. y , y x, about y -axis
x
2
1 5
2
y 2 y 5 y 2 0 y 0.5, 2
y 2
V
2
0.5
2.5 y
2
2
2
1
dy 6.25 5 y y 2 y 2 dy
0.5
y
2
25
5y
y
1
y
2
3 y1
4
2
3
2
25
8 1 25 5 1
10 2
3 2 8 8 24
2
75 60 16 3 75 15 1 48
136 109 9
6
24
24
8
Homework, Page 398
Find the volume of the solid obtained by rotating the region
enclosed by the graphs about the given axis.
1
3
43. y x , y x 3 , about y -axis
1
x y ,x y y
3
3
V
1
1
y
1
3
2
y
1
3
y 3 y y 9 y 1, 0 1
3 2
dy
1
1
y
2
3
y 6 dy
1
y 3 y7
3 1 3 1 1
7
5
7
5
7
5
3
1
5
32
21 5 21 5 32
V
35 35
35
35
Homework, Page 398
Find the volume of the solid obtained by rotating the region
enclosed by the graphs about the given axis.
45. y e x , y 1 e x , x 0, about y 4
e x 1 e x 2e x 1 e x 0.5 x 0.69314718 A
V
A
0
4 1 4 e
V 6.748
x
2
x 2
dx 2.148 6.748
Homework, Page 398
Find the volume of the solid obtained by rotating the region
enclosed by the graphs about the given axis.
47. y 2 4 x, y x, y 0, about x-axis
y 2 x 2 x x 4 x x 2 x 2 4 x 0 x 0, 4
4
V 2 x
0
2
4
2 x
x dx 4 x x dx 2 x
0
3 0
2
4
64 32
32
32
V
3
3
3
3
2
Homework,
Page
398
2/3
2/3
49.
Sketch the hypocycloid x + y = 1 and find the volume
of the solid obtained by revolving it about the x-axis.
y
2
3
1 x
V
1
1
2
3
y 1 x
dx 1 x dx
y = 1/(x+0.62)-0.62
y 1 x
1 x
2
2
3
3
2
3
2
3
2
4
0
0.305 0.957 V 0.957
2
3
3
3
3
Homework, Page 398
51.
A bead is formed by removing a cylinder of radius r from
the center of a sphere of radius R. (Figure 12) Find the volume of
the bead with r = 1 and R = 2.
2
x 2 y 2 4 1 y 2 4
y 3 x2 4 y 2
V
3
3
4 y 1 dy
y
3y
3
3
V 4 3
2
2
3
3
3
3 3 3 3 3
4
3
Jon Rogawski
Calculus, ET
First Edition
Chapter 7: Techniques of Integration
Section 7.1: Numerical Integration
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
The shaded area in Figure 1 cannot be calculated directly using a x2
definite integral, since there is not an explicit antiderivative for e 2
Instead, we will rely on numerical approximation using the
trapezoidal method
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
If we divide the interval [a, b] into N even intervals, the area may be
found using the Trapezoidal Rule
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
As shown in Figure 3,
the area of the
trapezoidal segment
is equal to the average
of the left- and rightRAM areas.
As shown in table one, by increasing
the size of N, we can attain whatever
degree of accuracy we may need.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 5 illustrates how a mid point estimate rectangle has the same
area as a trapezoid where the top of the trapezoid is tangent to the
curve at the midpoint of the interval.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
Calculate TN and MN for the value of N indicated.
4
2. 0 xdx
N 4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
Calculate TN and MN for the value of N indicated.
2
8. 1 ln xdx
N 5
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
Calculate the approximation to the volume of the solid obtained
by rotating the graph about the .
23. y cos x; 0, ; x-axis; M 8
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
We frequently are concerned with the accuracy of the estimate
obtained using either the trapezoidal or midpoint method. They
may be defined as follows:
Error TN TN a f x dx
b
Error M N M N a f x dx
b
If we assume f ″ (x) exists and is continuous on our interval, we
may use Theorem 1.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 6 shows how trapezoidal estimates for areas under curves
are more accurate for those with small values of f ″ .
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 6 shows the points we would use in calculating T6 and M6
for an approximation to the area of the shaded region in Figure 8.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 10 illustrates how trapezoids provide an underestimate of
areas under concave down curves and midpoints provide overestimates. The opposite holds true for concave up curves.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
State whether TN or MN overestimates or underestimates the integral
and find a bound for the error. Do not calculate for TN or MN.
2
32. 1 ln xdx
M10
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
Use the Error Bound to find a value of N for which the
Error (TN) ≤ 10 – 6.
5 dx
36. 2
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework
Homework Assignment #16
Read Section 7.2
Page 424, Exercises: 1 – 11(Odd), 25, 29,
33, 37
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company