Homework



Homework Assignment #17
Read Section 7.2
Page 424, Exercises: 1 – 11(Odd), 25, 29,
33, 37
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
Calculate TN and MN for the value of N indicated.
2
1. 0 x 2 dx
N 4
x
2 1
 
4 2
x2
0
1
0
1
16
4
1
1
2
9
4

16
1
5
1 25
4
3
9
2
7
49
4
16
4
16
1 1
44
9
1
TN 
02
 2 1  2
4 
 2.75
4
4
2 2
16
1 1
84
MN 
 9  25  49

 2.625
16
16
16
16
2
32
  
4
3
2
4
  

Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
Calculate TN and MN for the value of N indicated.
4
3. 1 x3dx
N 6
3 1

6 2
x
1
x3
1
x
11
5
4
125
64
3
7
9
2
5
2
4
4
2
27
343
729
125
8
8
64
64
8
13
7
15
4
4
2
4
2197
343
3375
64
64
8
64
3
4
x 3 1331
27
64
1 1
TN 
1  2 27  2  8   2 125  2  27   2 343  64
8
8
8
2 2
2070

 64.6875
32
  





Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
Calculate TN and MN for the value of N indicated.
4
3. 1 x3dx
N 6
3 1

6 2
x
1
x3
1
x
11
5
4
125
64
3
7
2
9
5
2
4
4
2
27
343
729
125
8
8
64
64
8
13
7
15
4
4
2
4
2197
343
3375
64
64
8
64
3
4
x 3 1331
27
64
1 125
MN 
 343  729  1331  2197  3375
64
64
64
64
64
64
2
8100

 63.28125
128


Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
Calculate TN and MN for the value of N indicated.
4 dx
5. 1
N 6
x
3 1

6 2
x
1 5
1
1 4
x
4
5
3
2
2
3
7
4
4
7
2
9
1
4
2
4
9
5
2
2
5
11
4
4
11
3
13
1
4
3
4
13
          
7
2
2
7

15
4
4
15
4
1
4
1 1
TN 
1  2 2  2 1  2 2  2 1  2 2  1  1.405
3
2
5
3
7
4
2 2
1 4
MN 
4 4 4 4 4
 1.377
5
7
9
11
13
15
2


Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
Calculate TN and MN for the value of N indicated.

7. 0 4 sec xdx
N 6

0

4
6

24
x
0
sec x
1
x
7




3
5
48
24
16
1.002 1.009 1.020
48
6
16
24
sec x 1.115 1.155 1.203 1.260

12
1.035
11
5

48
8
1.056 1.082

48
4
1.330 1.414
1  1  2 1.009   2 1.035   2 1.082   13.496
TN 
 0.883

 
2 24  2 1.155   2 1.260   1.414
48

 1.002  1.020  1.056  1.115  6.726
MN  

 0.880

24  1.203  1.330
24

Example, Page 424
Calculate TN and MN for the value of N indicated.
2
9. 1 ln xdx
N 5
2 1
 0.2
5
x
1
1.1
1.2
1.3
1.4
1.5
ln x
0
x
1.6
ln x 0.47
0.095 0.182 0.262 0.336 0.405
1.7
0.53
1.8
1.9
2
0.588 0.642 0.693
1 1  0  2  0.182   2  0.336   2  0.47   2  0.588  
TN 


2 5  0.693

 0.38454
MN 
1
 0.095  0.262  0.405  0.53  0.642   0.3868
5
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
Calculate xTN and MN for the value of N indicated.
2 e
11. 0
dx
N 8
x 1
20 1
  0.25
8
4
x
0
0.125
f  x
1
x
1.125
0.25
0.375
0.5
0.625
0.75
0.875
1
1.007 1.027 1.058 1.099 1.150 1.210 1.279 1.359
1.25
1.375
1.5
1.625
1.75
1.875
2
f  x  1.450 1.551 1.665 1.793 1.934 2.093 2.268 2.463
1 1 1  2 1.027   2 1.099   2 1.210   2 1.359  
TN 

  2.966
2 4  2 1.551  2 1.793  2  2.093   2.463

1 1.007  1.058  1.150  1.279  1.450  1.665 
MN  
 2.953

4  1.934  2.268

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 .
 x2
25. y  e ;  0,1; x-axis; T8
1 0 1

8
8
x
0
0.125 0.25 0.375 0.5
 
 e
 x2
2
3.142 3.045 2.772 2.371 1.906
x
 
 e
 x2
0.625
0.75
0.875
1
2
1.438 1.020 0.679 0.425
1 1  3.142  2  3.045   2  2.772   2  2.371  2 1.906  
T8 


2 8  2 1.438   2 1.020   2  0.679   0.425

Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
 1.877
Example, Page 424
29.

 0
2
6
x
Calculate M6 for the integral I  0 2 cos xdx .



12
3
5
7
9
24
24
24
24
24
cos x 0.991 0.924 0.793 0.609 0.383
M6 

11
24
0.131
 0.991  0.924  0.793  0.609  0.383  0.131
12
 1.003
(a)
Is M6 too large or too small? Explain graphically.
M6 is too large,
since the curve is
concave down.
y

x

Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
29.
(b)
Show that K2 = 1 may be used in the Error Bound
and find the bound of the error.
f  cos x, f    sin x, f    cos x
 
 


f  0    cos  0   1, f     cos    0
2
2
f   x   1  Error  M N  
Error  M 6  

1  2  0
24  6 
2

K2 b  a 
3

3
24 N 2
3
8  0.0045
864
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 424
29.
(c)
Evaluate I and check that the actual error is less
than the bound computed in (b)


I  0 2 cos xdx  sin x 0 2  1
1.003  1  0.003  0.0045
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.

33. 0 4 cos xdx
M 20
M 10 overestimates the integral.
f   x    sin x, f   x    cos x
 
f   0   1, f  
2

4
2
f   x   1  Error  M N  
Error  M 10  

1  4  0
24 10 
2

K2 b  a 
3
24 N 2
3

3
64  0.0002018
2400
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.
3
37. 0 e x dx
f   x   e  x , f   x   e  x
f   0   1, f   3  e 3  0.0498
f   x   1  Error TN  
6
10 
1 3  0 
12 N 2
3
K2 b  a 
3
12 N 2
27
27
2

N 
2
12 N
12 106
N  2.25 106  N  1500
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Jon Rogawski
Calculus, ET
First Edition
Chapter 7: Techniques of Integration
Section 7.2: Integration by Parts
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Derivation of Integration by Parts
Formula
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
The Integration by Parts formula is given in the following box:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 431
Evaluate the integral using Integration by Parts and the given u
& v′.
2.  xe2 x dx, u  x, v  e2 x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 431
Use Integration by Parts to evaluate the integral.
ln x
16.  2 dx
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 431
Use Integration by Parts to evaluate the integral.
20.  x 2 ln xdx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 431
Use Integration by Parts, substitution or both, if necessary.
42.  xe x dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company