Chapter 7
Integral Calculus
The basic concepts of differential calculus were covered in the preceding chapter.
This chapter will be devoted to integral
calculus, which is the other broad area of calculus. The next chapter will be devoted to how both differential and integral calculus manipulations can be performed with MATLAB.
1

Anti-Derivatives
An anti-derivative of a function f(x) is a new function F(x) such that dx

2

Indefinite and Definite Integrals
Indefinite

Definite
 x
1 x
2
3

Definite Integral as Area Under the Curve y y
K y
1 y
2 y
3 y
4 y
5 a b
Approximate Area
  k y k
 x x
4

Exact Area as Definite Integral
 a b ydx
 lim x dx
 k y k
 x
5

Definite Integral with Variable Upper
Limit
 a x ydx
More “proper” form with “dummy” variable
 a x
6

Area Under a Straight-Line Segment y

( ) y
2
L y
1
 x
1 x
2 ydx
 x
1 x
2
1
2
 y
2
 y
1
 x
2
 x
1
 x
7

Example 7-1. Determine

0
12 ydx y

( )
20 20
12 4 6
8
9
2
-10
-20 x
8

Example 7-1. Continuation.
y

( )
20 20
9 12 4 6
8
2
-10
-20

0
12 ydx

40 0 20 20 0 30 10 x
9

Example 7-2. Determine

0 x ydx y

( )
20 20
12 4 6
8
9
2
-10
-20 x
10

Guidelines
1. If y is a non-zero constant, integral is either increasing or decreasing linearly.
2. If segment is triangular, integral is increasing or decreasing as a parabola.
3. If y=0, integral remains at previous level.
4. Integral moves up or down from previous level; i.e., no sudden jumps.
5. Beginning and end points are good reference levels.
11

(a) y

( )
20
4 6
2
(b)

0 x   x ydx f x dx
0
( )
40
-20
20
8
9
40
-10
12 x
20
2 4 6 8 9
10
12 x
12

Tabulation of Integrals
 
I
  a b
I

( )
 b a

( )

( )
13

Table 7-1. Common Integrals.
  Integral Number
I-1
I-2 ( )

( ) x n a
 n
 
1
 e ax

( )
 
( ) ax x n

1 n

1 e ax a
I-3
I-4
I-5
1 x sin ax ln x
I-6 cos ax sin
2 ax

1 a cos ax
1 sin ax a
1 x

2 4
1 a sin 2 ax
I-7
I-8
I-9
14

2 cos ax x sin ax x cos ax sin ax cos bx for a
2  b
2 xe ax ln x
1 ax
2
 b
Table 7-1. Continuation.
sin ax cos ax
1 x

2 4
1 a sin 2 ax
1 a
2 sin ax
 x a cos ax
1 cos ax
 x a sin ax a
2

1
2( a
 b )
2 cos( a

2 a
) sin
 ax cos( a
 b x
2( a
 b ) e ax
 ax

1
 a
2 x
 ln x

1

1 ab tan

1

 x a b


I-10
I-11
I-12
I-13
I-14
I-15
I-16
I-17
15

In Examples 7-3 through 7-5 that follow, determine the following integral in each case: z
  ydx
16

Example 7-3 y

12 e
4 x z
 
12
4 x e dx

12 e
4 x
4

3 e
4 x 
C

C
17

Example 7-4 y

12 sin 2 x z
 
12 sin 2 xdx


12 


1
(2)
2 sin 2 x

3sin 2 x

6 cos 2 x
2 cos 2 x


C


18

Example 7-5 z
  y

6 x
2 



6 x
2 
3 x
3 x


 dx
 
6
2 x dx
  3 x dx
6 x
3
 
3
3ln

2 x
3 
3ln
19

In Examples 7-6 and 7-7 that follow, determine the definite integral in each case as defined below.
  b
I ydx a
20

Example 7-6
I
 
0
 sin xdx
I
 
0
 sin
  cos
 xdx
  cos
 cos 0
 x
 
0
( 1) ( 1) 2
21

Example 7-7
I
 
0
1
8

2 x xe dx
I
 
0
1
8

2 x xe dx


8 e

2 x
(2)
2
2 e

2
  
1

1
0
  
2 e

0
  
 
6 e

2
2 1.188
22

Displacement, Velocity, and Acceleration a
 a t
 2 2
( ) acceleration in meters/second (m/s ) v

( )
 velocity in meters/second (m/s) y

( )
   dv

 dv
 dt dv

 dt
 dt

( )
 dv
 
( )
 dv
 v
23

Displacement, Velocity, and Acceleration
Continuation dy dt
 v
 
( )

C
1




( )

 y
 
( )

C
2
24

Alternate Formulation in Terms of
Definite Integrals
 
0 t
( )
 v (0)
 
0 t
( )
 y (0)
25

Example 7-8. An object experiences acceleration as given by
( )

20 e

2 t v (0)

0
(0)

0
Determine the velocity and displacement.
26

Example 7-8. Continuation.
dv

( )

20 e

2 t dt
( )
 
20

2 t e dt

20

2 e

2 t 
C
1
 
10 e

2 t 
C
1 v (0)
 
10 e

0 
C
1
10 C
1

C

1
10
  e

2 t
0
27

Example 7-8. Continuation.
( )
  
 e

2 t
 dt

10 t

10

2 e

2 t 
C
2

10 t

5 e

2 t 
C
2 y (0)
  e

0 
C
2
5 C
2

0
C
 
2
5 y t
 t

5 e
 t 
5
28

Example 7-9. Rework previous example using definite integral forms.
 
0 t
( )
 
10 e

2 t
 v (0)
 
0 t
20

2 t e dt

10 e

0

  e

2 t

20 e

2 t
2

 t
0 t
0
29

Example 7-10. Plot the three functions of the preceding examples.
( )

20 e

2 t
  e

2 t
( ) 10 t

5 e

2 t 
5
30

Example 7-10. Continuation.
>> t = 0:0.02:2;
>> a = 20*exp(-2*t);
>> v = 10 -10*exp(-2*t);
>> y = 10*t + 5*exp(-2*t) - 5;
>> plot(t, a, t, v, t, y)
The plots are shown on the next slide.
31

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