Homework



Homework Assignment #2
Read Section 5.3
Page 321, Exercises: 1 – 61(EOO), 71, skip
41
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Draw a graph of the signed area represented by the integral and
compute it using geometry.
3
1. 3 2xdx
y






x
1
1
 2 xdx    3 6    3 6   0
2
2
3
3



















Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

Example, Page 321
Draw a graph of the signed area represented by the integral and
compute it using geometry.
8
5. 6  7  x  dx
y






1
1
  7  x  dx  1 1  1 1  0
2
2
8
6
x



















Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Draw a graph of the signed area represented by the integral and
compute it using geometry.
2
9. 2  2  x  dx
y



1
  2  x  dx   4 2   4
2
2
2

x












Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company

Example, Page 321
13.
a)
b)
c)
d)
Evaluate the integrals for f (x) shown in Figure 14.
1

2
f
x
dx



1



  
2
2
1
1
3
2
2
6
0 f  x  dx    1    2  
2
2
2
1
1
3
2
2
4
1 f  x  dx    1    2  
4
4
4
1
1
7
2
2
6
1 f  x  dx    1    2  
4
2
4
2
0
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Sketch the signed area represented by the integral. Indicate the
regions of positive and negative area.
2
17 0  x  x 2  dx
y




x

+



Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Sketch the signed area represented by the integral. Indicate the
regions of positive and negative area.
2
21.  1 ln xdx
2
y


+

-


x



Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Determine the sign of the integral without calculating it. Draw a
graph if necessary.
2
25. 0 x sin xdx
y


The sign of the integral
is negative.
x













Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Calculate the Riemann sum for the given function, partition, and
choice of intermediate points. Also, sketch the graph of f and the
rectangles corresponding to R (f, P, C).
29. f  x   x  1, P  2, 1.6, 1.2, 0.8, 0.4, 0
C  1.7, 1.3, 0.9, 0.5, 0
y

f  1.7   0.7, f  1.3  0.3,
f  0.9   0.1, f  0.5   0.5,
x



f  0   1.0
0
2

 x  1 dx  0.4  0.7  0.3  0.1  0.5  1.0 

 0.24
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Use the basic properties of the integral and the formulas in the
summary to calculate the integral.
3
33. 0  3t  4  dt
  3t  4  dt  0 3tdt  0 4dt  30 tdt  40 dt
3
0
3
3
3
3
27
1 2
 3   3   4  3 
 12  25.5
2
2

Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Use the basic properties of the integral and the formulas in the
summary to calculate the integral.
1
37.  a  x 2  x  dx
2
2
2
  x  x  dx   a  x  x  dx  0  x  x  dx
1
a
0
1
a
2
2
x

x
dx

x

 0   x  dx
a
 x  dx    x  dx    x  dx    x  dx
  0
  0
1
2
a
0
1
0
2
1
0
1
1
1 3 1 2
3
2
    a     a   1  1
3
2
3
2
a3 a 2 5
  
3 2 6
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Use the formulas in the summary and Equation 9 to evaluate the
integral.
2
45. 0  x  x 3  dx
 2
1
2
  x  x  dx    x  dx    x  dx   2  
2
4
 2  4  2
2
0
3
2
0
2
0
3
4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Use the formulas in the summary and Equation 9 to evaluate the
integral.
2
49. 1  x  x 3  dx

3
3
3
  x  x  dx  0  x  dx  0  x  dx  0  x  dx  0  x  dx
2
1
2
2
1
1
 0  x  dx  0  x 3  dx  0  x  dx  0  x 3  dx
2
2
1
 2  1 2 1
1
2
  2 
 1 
2
4
2
4
1 1
9
 24   
2 4
4
4

1
4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Calculate the integral, assuming
5
5
f
x
dx

5,


0
0 g  x  dx  12
53.
  3 f  x   5 g  x   dx
5
0
  3 f  x   5 g  x   dx   3 f  x  dx   5 g  x  dx
5
0
5
0
5
0
 30 f  x  dx  50 g  x  dx
5
5
 3  5   5 12   15  60  45
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Calculate the integral, assuming
1
2
4
f
x
dx

1,
f
x
dx

4,




0
0
1 f  x  dx  7
57.
 f  x  dx
1
4
 f  x  dx   1 f  x  dx  7
1
4
4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Express each integral as a single integral.
9
5
61. 2 f  x  dx  2 f  x  dx
 f  x  dx  2 f  x  dx  5 f  x  dx
9
2
5
9
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Calculate the integral, assuming f is an integrable function such .
b
that 1 f  x  dx  1  b 1 for all b  0.
4
65. 1  4 f  x   2  dx
  4 f  x   2  dx  4 1 f  x  dx  2 1 dx
4
1
4
4
 4 1  41   2  4  1
3
 4    6  3
4
  4 f  x   2  dx  3
4
1
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 321
Calculate the integral.
6
71. 0 3  x dx
y


1
1
3
3

   3 3
2
2
9
 3  x dx 
6
0


x













Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Jon Rogawski
Calculus, ET
First Edition
Chapter 5: The Integral
Section 5.3: The Fundamental Theorem
of Calculus, Part I
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
The Fundamental Theorem of Calculus ties together the operations of
differentiation and integration in a relationship. Specifically, the
integral of f (x) evaluated from a to b is equal to the antiderivative of
f (x) evaluated at x = b minus the antiderivative of f (x) evaluated at
x = a.
When evaluating a definite integral, we frequently have
an intermediate step as follows:
 f  x  dx  F  x  a  F  b   F  a 
b
b
a
where F  x  a indicates we are evaluating F  x  between a and b.
b
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Verify the assertion about the area under the curve in Figure 2.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Find the area under the curve between
the limits shown in figure 3.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Verify that the shaded
are in Figure 4 is 2.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Calculate the shaded area in
Figure 5.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Recalling that the antiderivative of f (x) = x–1 is F(x) = ln x, the FTC
tells us:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Verify the relationship in the
caption to Figure 6.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Find the areas
of the shaded
regions in
Figure 7.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Basic Antiderivative Formulae
n 1
x
n
x
 dx  n  1  C
dx
 x  ln x  C
x
x
e
dx

e
C

 sin xdx   cos x  C
 cos xdx  sin x  C
2
sec
 xdx  tan x  C
2
csc
 xdx   cot x  C
 sec x tan xdx  sec x  C
 csc x cot xdx   csc x  C
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 329
Evaluate the integral using the FTC 1.
4 dx
12. 12
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 329
Evaluate the integral using the FTC 1.
4
16. 0  3 x 5  x 2  2 x  dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 329
Evaluate the integral using the FTC 1.
4
22. 0 ydy
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 329
Evaluate the integral using the FTC 1.
4
30. 2  2 dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 329
Evaluate the integral using the FTC 1.
2
32. 0 cos  d
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 329
Write the integral as a sum of integrals without the absolute
values and evaluate.
5
42. 0 3  x dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 329
Evaluate the integral in terms of the constants.
a
48. b x 4 dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework




Homework Assignment #3
Read Section 5.4
Page 329, Exercises: 1 – 49(EOO)
Quiz next time
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company