5.d – Applications of
Integrals
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Definite Integrals and Area
The definite integral is related to the area bound by the
function f(x), the x-axis, and the lines x = a and x = b.
definite integrals do not always yield area since we
know that definite integrals can give negative values.
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Indefinite Integrals and Area
Examples: Compute the definite integrals using your graphing
calculators. Then compute the area bound by the graphs of the
integrands, the x-axis, and x = a and x = b. In which cases do
definite integrals yield actual area? Does definite integrals always
yield actual area?
 a  2
2
 b 1  3  2 x  dx
3
4  x dx
2
(d ) Evaluate:
7
1
x
2
 c  0
10
x  5 dx
 6 x  5 dx
Does the value of (d) represent the actual area bound by x2 – 6x + 5,
the x-axis, x = -1 and x = 7? Write two examples of definite integrals
that will yield the actual area.
3
More Properties of the Definite Integrals
1.

b
a
c dx  c  b  a 
2. If f (x) ≥ 0 for a ≤ x≤ b, then a f  x  dx  0
b
3. If f (x) ≥ g (x) for a ≤ x ≤ b, then
 f  x  dx   g  x  dx
b
b
a
a
4. If m ≤ f (x) ≤ M for a ≤ x ≤ b, then
m  b  a    f  x  dx  M  b  a 
b
a
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Examples
1. Use the properties of integrals to verify the
inequality without evaluating the integrals.
 a  1
2
5  x dx  
2
1
x  1 dx
b

c
Estimate   x3  3x  3 dx a previous property.
6

 /2
 /6
sin x dx 

3
2
0
5
The Net Change Theorem
The integral of a rate of change f   x  is the net change:

b
a
f '( x) dx  f (b)  f (a) (1)
Must Be A Rate Of Change
Important: For the net change theorem to apply, the
integrand 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 (x) from a to
b.
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Examples
2. What does the integral below represent if v(t)
is the velocity of a particle in m/s.

11
5
v(t ) dt
3. A honeybee population starts with 100 bees
and increases at a rate of n(t). What does
15
100   n  t  dt represent?
0
7
Examples
4. 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?
5. 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?
8
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
 v t 
b
a
dt  total
distance traveled
______________
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Examples
6. 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 displacement during the given time
interval. (c) The total distance traveled during
the time interval.
a t   2t  3, v  0  4, 0  t  3
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Examples
7. Water flows from the bottom of a storage tank
at a rate of r(t) = 200 – 4t liters per minute,
where 0 ≤ t ≤ 50. (a) Find the amount of water
that flows from the tank in the first 10
minutes. (b) How many liters of water were in
the tank?
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