homework homework assignment #47 read section 7.1 page 398, exercises: 23 – 51(odd) rogawski...

Homework

Homework Assignment #47 Read Section 7.1 Page 398, Exercises: 23 – 51(Odd)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 398Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

23. -axisx

2 22 2

0

2 4 2

0

25 3

0

6 2

36 4 4

32 32 32 4 64 0

5 3 5 3

960 96 160 704 704

15 15 15

V x dx

x x dx

x xx

V

Homework, Page 398Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

25. 2y

22 2 2

0

2 4

0

25

0

2 6 2 2

16

32 16 32 0

5 5

160 32 128 128

5 5 5

V x dx

x dx

xx

V

Homework, Page 398Find the volume of the solid obtained by rotating region A in Figure 10 about the given axis.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

27. 3x

2 2

26 2

2

6

2

63

2 2

2

2 2 2

2 3 0 3

2 3 2 9 9

2 2 3

322

18 12 2 8 2 4 0 24 24

y x y x x y

V y dy

y y dy

yyy

V

Homework, Page 398Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

29. -axisx

2 222 4 2

0 0

25 3

0

2 4 4

4 45 3

32 32 8 0

5 3

96 160 120 376 376

15 15 15

V x dx x x dx

x xx

V

Homework, Page 398Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

31. 6y

22 22 2 4 2

0 0

25 3

0

6 0 6 2 36 8 16

32 64 20 8 40 0

5 3 5 3

600 96 320 824 824

15 15 15

V x dx x x dx

x xx

V

Homework, Page 398Find the volume of the solid obtained by rotating region B in Figure 10 about the given axis.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

33. 2x

2 2

262

2

63

226

2

2

2 2 2

2 2 2 2

2 8 4 4 2 2 8 2 4

3 22

8 8 8 12 8 18 4 0 2

3 3

72 64 32 32 8

3 3 3

y x x y x y

V y dy

y yy y dy y

V

Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

235. , 12 , 0, about 2y x y x x y

2 2

3 22 2

0

3 2 4 2

0

35

3 2 4 2 3

00

12 12 0 4 3 0 4,3

12 2 2

196 28 4 4

192 28 3 192 145

243 2880 630 135 243 576 126 27

5 5

1872 1872

5

x x x x x x x

V x x dx

x x x x dx

xx x x dx x x x

V

5

Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

37. 16 , 3 12, 0, about -axisy x y x x y

2

15 16 2

12 15

16 3 12 4 4 1 15

4 163

x x x x y

yV dy y dy

Homework, Page 398

215 16 2

12 15

215 16 2

12 15

15 163 3

2 2

12 15

37. Continued.

4 163

8 16 256 32

9 3

4 16 256 16

27 3 3

3375 1728 300 240 192 192

27 27

yV dy y dy

yy dy y y dy

y yy y y y

4096 3375 4096 4096 3840 3600

3 3

3375 1620 1728 4096 720 3375 27 9 4

27 3 27 3

Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

22

939. , 10 , about -axisy y x x

x

2 4 22

23 322 2 4 4

21 1

910 10 9 0 1, 3

910 100 20 81

x x x xx

V x dx x x x dxx

Homework, Page 398

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2

3 322 2 4 421 1

3 33 5 3 3 5

3

1 1

39. Continued.

910 100 20 81

20 27 100 20 81 100

3 5 3 3 5

243 20 1 300 180 1 100 27

5 3 5

600 243 5 1500 1

5

V x dx x x x dxx

x x x x xx x

x

00 3 405 2544 1808 736

15 15 15

1472Two regions enclosed by the curves, means two volumes

15V

Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

1 541. , , about -axis

2y y x y

x

2

22 22 2 2

0.5 0.5

22 3

12

1 52 5 2 0 0.5,2

2

12.5 6.25 5

25 5 1

4 2 3

25 8 1 25 5 1 10 2

2 3 2 8 8 24

75 60 16 3 75 15 1 48 136 109

6 24 24

y y y yy

V y dy y y y dyy

y yy

y

9

8

Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

13 343. , , about -axisy x y x y

1 13 3 93 3

21 11 223 63 3

1 1

15 73

1

, 1,0 1

3 13 1 1

5 7 5 7 5 73

21 5 21 5 32 32

35 35 35 35

x y x y y y y y y

V y y dy y y dy

y y

V

Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

45. , 1 , 0, about 4x xy e y e x y

2 2

0

1 2 1 0.5 0.69314718

4 1 4 2.148 6.748

6.748

x x x x

A x x

e e e e x A

V e dx

V

Homework, Page 398Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.

247. 4 , , 0, about -axisy x y x y x

2 2

4324 42 2 2

0 00

2 2 4 4 0 0,4

2 4 23

64 32 32 32

3 3 3

y x x x x x x x x

xV x x dx x x dx x

V

Homework, Page 39849. Sketch the hypocycloid x 2/3 + y 2/3 = 1 and find the volume of the solid obtained by revolving it about the x-axis.

3 32 2 2 223 3 3 3

23 31 42 2

3 3

1 0

1 1 1

1 1

0.305 0.957 0.957

y x y x y x

V x dx x dx

V

y = 1/(x+0.62)-0.62

Homework, Page 39851. 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.

22 2 2

2 2

3 22

3

33

3

4 1 4

3 4

4 1

3 3 3 3 3 3 3 4 33

4 3

x y y

y x y

V y dy

yy

V

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Chapter 7: Techniques of IntegrationSection 7.1: Numerical Integration

Jon Rogawski

Calculus, ETFirst Edition

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

The shaded area in Figure 1 cannot be calculated directly using a definite integral, since there is not an explicit antiderivative forInstead, we will rely on numerical approximation using the trapezoidal method

2

2x

e

Rogawski CalculusCopyright © 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 CalculusCopyright © 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 right-RAM areas.

As shown in table one, by increasingthe size of N, we can attain whateverdegree of accuracy we may need.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Figure 5 illustrates how a mid point estimate rectangle has the samearea as a trapezoid where the top of the trapezoid is tangent to the curve at the midpoint of the interval.

Example, Page 424Calculate TN and MN for the value of N indicated.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

4

02. 4xdx N

Example, Page 424Calculate TN and MN for the value of N indicated.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

2

18. ln 5xdx N

Example, Page 424Calculate the approximation to the volume of the solid obtained by rotating the graph about the .

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

823. cos ; 0, ; -axis; 2

y x x M

Rogawski CalculusCopyright © 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 Errorb bN N N Na aT T f x dx M M f x dx

If we assume f ″ (x) exists and is continuous on our interval, we may use Theorem 1.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Figure 6 shows how trapezoidal estimates for areas under curvesare more accurate for those with small values of f ″ .

Rogawski CalculusCopyright © 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 CalculusCopyright © 2008 W. H. Freeman and Company

Figure 10 illustrates how trapezoids provide an underestimate of areas under concave down curves and midpoints provide over-estimates. The opposite holds true for concave up curves.

Example, Page 424State whether TN or MN overestimates or underestimates the integral and find a bound for the error. Do not calculate for TN or MN.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

210132. ln xdx M

Example, Page 424Use the Error Bound to find a value of N for which the Error (TN) ≤ 10 – 6.

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

5

236. dx

x

Homework

Homework Assignment #16 Read Section 7.2 Page 424, Exercises: 1 – 11(Odd), 25, 29,

33, 37

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company