1

5.4 – Indefinite Integrals and The Net Change Theorem

2

Indefinite Integrals

Use WolframAlpha to determine the following.

integral ( ),f x dx f x x

2. cos .a x dx b x dx

Question: What does represent? f x dx

3

Indefinite Integrals

In other words, F(x) is the _________________ of f (x).

f x dx F x means F x f x

4

Examples

Evaluate each indefinite integral.

21/4

31. 3 ln 3xx dx

x

2

2

52. 3 sec

1

ue u duu

sin 23.

sin

ydy

y

5

Definite Integrals

where F(x) is the general antiderivative of f (x).

b b b

a aaf x dx F x F x

6

Examples

5

02. 2 4cosxe x dx

4

01. 2 5 3 1v v dv

9

1

3 23.

xdx

x

2/3

20

sin sin tan4.

secd

3 /2

05. sin x dx

7

The Net Change Theorem

The integral of a rate of change is the net change: F x

( ) ( )a

bF x dx F b F a

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.

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.

(1)

8

Examples

1. The current in a wire, I, is defined as the derivative of the charge, Q. That, isI(t) = Q(t). What does represent?

b

aI t dt

2. A honeybee population starts with 100 bees and increases at a rate of n(t). What does represent?

15

0100 n t dt

9

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 represent?

5

3f x dx

4. If the units for x are feet and the units for a(x) are pounds per foot, what are the units for da/dx. What units does have?

8

2a x dx

10

Example

A particle moves with a velocity v(t). What does and represent?

( )b

av t dt

b

av t dt

total distance traveledb

av t dt

|

0

s(t)

displacementb

av t dt

t = a●

●t = b

11

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.

2 3, 0 4, 0 3a t t v t

12

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.