5.4 – Indefinite Integrals and
The Net Change Theorem
1
Indefinite Integrals
Use WolframAlpha to determine the following.
 f  x  dx  integral  f ( x), x
a.  cos  x  dx
Question: What does
b.

2
x dx
 f  x  dx represent?
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Indefinite Integrals
 f  x  dx  F  x  means F   x   f  x 
In other words, F(x) is the _________________
of f (x).
3
Examples
Evaluate each indefinite integral.
3
 2

x
1.   x  1/4  3 ln  3  dx
x


 u
5
2
2.   3e  sec u 
2
1

u


 du

sin  2 y 
3. 
dy
sin  y 
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Definite Integrals
 f  x  dx   F  x 
b
b
a
a
 F  x a
b
where F(x) is the general antiderivative of f (x).
5
Examples
1.   2v  5 3v  1 dv
4
0
2.   2e  4 cos x  dx
5
x
0
3. 
9
1
4. 
 /3
0
5. 
sin   sin  tan 2 
d
2
sec 
3 /2
0
sin x dx
3x  2
dx
x
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The Net Change Theorem
The integral of a rate of change F   x  is the net change:

a
b
F   x  dx  F (b)  F (a )
(1)
Must Be A Rate Of Change
Important: For the net change theorem to apply, the
function in the integral must be a rate of change.
Meaning: If F (x) represents a rate of change (m/sec),
then (1) above represents the net change in F (m) from a
to b.
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Examples
1. The current in a wire, I, is defined as the
derivative of the charge, Q. That, is
b
I(t) = Q(t). What does  a I  t  dt represent?
2. A honeybee population starts with 100 bees
and increases at a rate of n(t). What does
15
100   n  t  dt represent?
0
8
Examples
3. If f (x) is the slope of a trail at a distance of x
miles from the start of the trail, what does
5
3 f  x  dx represent?
4. If the units for x are feet and the units for a(x)
are pounds per foot, what8 are the units for
da/dx. What units does  2 a  x  dx have?
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Example
A particle moves with a velocity v(t). What does
b
b
 a v(t ) dt and  a v  t  dt represent?
t=a
●
●
t=b
|
s(t)
0
 v  t  dt  displacement
b
a

b
a
v  t  dt  total distance traveled
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Examples
1. The acceleration functions (in m/s2) and the
initial velocity are given for a particle moving
along a line. Find (a) the velocity at time t and
(b) the distance traveled during the given time
interval.
a  t   2t  3, v  0  4, 0  t  3
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Examples
2. Water flows from the bottom of a storage tank
at a rate of r(t) = 200 – 4t liters per minute,
where 0 ≤ t ≤ 50. Find the amount of water
that flows from the tank in the first 10
minutes.
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