HCMUT – DEPARTMEND OF MATH. APPLIED
---------------------------------------------------------------------------------------------------------------------
CALCULUS FOR BUSINESS – 212
CHAPTER 4: INTEGRAL
•
PhD. NGUYỄN QUỐC LÂN (April, 2022)
CONTENTS
-----------------------------------------------------------------------------------------------------------------------------------
1- INDEFINITE INTEGRAL. ECONOMIC EXAMPLES
2- DEFINITE INTEGRAL. NET CHANGE
3- CONSUMER’S & PRODUCER’S SURPLUS
ANTIDERIVATIVE OR INDEFINITE INTEGRAL
-------------------------------------------------------------------------------------------------------------------------------
Find g ( x ) satisfying g / ( x ) = f ( x )  g ( x ) =  f ( x )dx : Antiderivative
x n+1
 x dx = n + 1 + C
n
x
x
e
dx
=
e
+C


dx
 x = ln x + C
mx
e
mx
e
 dx = m + C
1
f (ax + b )dx = F (ax + b ) + C
a
Example:
 2 x 3 − 7e 2 x + 5 + 3 dx

 
x

 2 x + 3 + 5 dx

 
3x − 1 
ECONOMIC EXAMPLE
-------------------------------------------------------------------------------------------------------------------------------
Example 5.1.5 (Chapter 5): A manufacturer has found that
marginal cost is 3q2 – 60q + 400 dollars/unit when q units have
been produced. The total cost of producing the first 2 units is
$900. What is the total cost of producing the first 5 units?
EXAMPLE 5.1.6 (HOFFMAN, CHAPTER 5)
-------------------------------------------------------------------------------------------------------------------------------
The population P(t) of a bacterial colony t hours after
observation begins is found to be changing at the rate 200e0.1t +
150e–0.03t. If the population was 200.000 bacteria when
observations began, what will the population be 12h later?
Derivative is the rate of change  Rate =
P / (t )
EXAMPLE 5.1.6 (HOFFMAN, CHAPTER 5)
-------------------------------------------------------------------------------------------------------------------------------
The population P(t) of a bacterial colony t hours after
observation begins is found to be changing at the rate 200e0.1t +
150e–0.03t. If the population was 200.000 bacteria when
observations began, what will the population be 12h later?
Derivative is the rate of change  Rate =
P / (t ) =
P / (t )
dP
= 200e 0.1t + 150e −0.03t
dt
(
)
 P(t ) =  200e 0.1t + 150e −0.03t dt = 2000e 0.1t − 5000e −0.03t + C
IC : P(0 ) = 200000  C = 203000  P(12 )  206.152
DEFINITE INTEGRAL
-------------------------------------------------------------------------------------------------------------------------------
For function y = f(x), x  [a, b] 
Divide [a, b] into small intervals with
points xi, introduce the integral sum &
take limit:
n
b
i =1
a
f ( xi )x =  f ( x )dx (Formal)

n→
A = lim
D: From x = a to x = b,
above Ox, under (C): y
= f(x)  0  Its Area
b
A =  f ( x ) dx
a
THE FUNDAMENTAL THEOREM OF CALCULUS
----------------------------------------------------------------------------------------------------------------------------------------------
I/ If f is continuous on [a, b] then the function g defined by
x
g ( x ) =  f (t )dt , a  x  b
a
is continuous on [a, b], differentiable on (a, b) & g’(x) = f(x)  So
g(x) is one antiderivative of f(x)
x
Example: Find the derivative of the function
g ( x ) =  1 + t 2 dt
0
g / (x) = 1 + x2
Solution: In fact, we just substitute
b
II/ Newton–Lebnitz formula:
 f (t )dt = F (b ) − F (a ) = F ( x )
a
b
a
NET CHANGE
-------------------------------------------------------------------------------------------------------------------------------
Previously, the rate of change Q’(x) of a quantity Q(x) is given
and the net change Q(b) – Q(a) in Q(x) when x varies from x = a
to x = b is required to find. Instead bof solving initial value
/
problem, we directly find: Q(b ) − Q(a ) =  Q ( x )dx
a
At a certain factory, the marginal cost is 3(q – 4)2 dolars/unit when
the level of production is q units. By how much the total
manufacturing cost increase if the level of production is raised
from 6 units to 10 units?
EXAMPLE 5.1.6 (HOFFMAN, CHAPTER 5)
-------------------------------------------------------------------------------------------------------------------------------
The population P(t) of a bacterial colony t hours after
observation begins is found to be changing at the rate 200e0.1t +
150e–0.03t. If the population was 200.000 bacteria when
observations began, what will the population be 12h later?
Derivative is the rate of change  Rate =
P / (t )
EXAMPLE
-------------------------------------------------------------------------------------------------------------------------------
A protein with mass m (grams) disintegrates into amino acids at a
rate given by:
dm
30
=− 2
dt
t +9
g/hr
What is the exact net change in mass of the protein during the
first 3 hours?
SUBSTITUTION RULE
-------------------------------------------------------------------------------------------------------------------------------
General  f u ( x )u / ( x )dx : u = u ( x )  du = u / ( x )dx  I =  f (u )du
Key: From the expression inside integral, recognize u = u(x).
2
2
(
)
f
x
+
C
xdx

u
=
x
+C

b
a
b
dx
 m 2 + x 2  x = m tan t
a
(mx + n )dx
mx + n
A
B
:
=
+
 ax 2 + bx + c ( x − x1 )( x − x2 ) x − x1 x − x2
INTEGRATION BY PART
-------------------------------------------------------------------------------------------------------------------------------
/
u
(
x
)
v
( x )dx =  udv :

P(x): polynomial 
1
Exam. :  xe2 x dx
0
u = u ( x )  du = u / dx
dv = v ( x )dx  v =  dv
/
 I = uv −  vdu
/

u
=
P
(
x
)

du
=
P
( x )dx

mx
 P( x )e dx : dv = e mx dx  v = e mx m

EXAMPLE
-------------------------------------------------------------------------------------------------------------------------------
A manufacturer find marginal cost
MC(q ) = (0.1q + 1)e0.03q
dollars/unit for q units. The total cost of producing 10 units is
$200. What is the total cost of producing the first 20 units?
SUBSTITUTION RULE
-------------------------------------------------------------------------------------------------------------------------------
General  f u ( x )u / ( x )dx : u = u ( x )  du = u / ( x )dx  I =  f (u )du
Key: From the expression inside integral, recognize u = u(x).
 (
b
)
f x 2 + C xdx  u = x 2 + C
a
INTEGRATION BY PART
-------------------------------------------------------------------------------------------------------------------------------
/
u
(
x
)
v
( x )dx =  udv :

P(x): polynomial 
1
Exam. :  xe2 x dx
0
u = u ( x )  du = u / dx
dv = v ( x )dx  v =  dv
/
 I = uv −  vdu
/

u
=
P
(
x
)

du
=
P
( x )dx

mx
 P( x )e dx : dv = e mx dx  v = e mx m

EXAMPLE
-------------------------------------------------------------------------------------------------------------------------------
A manufacturer find marginal cost
MC(q ) = (0.1q + 1)e0.03q
dollars/unit for q units. The total cost of producing 10 units is
$200. What is the total cost of producing the first 20 units?
CONSUMER’S SURPLUS
------------------------------------------------------------------------------------------------------------------------------------
Demand function P = f(Q)
& Q = Q0  P = P0. SOABC
= P0Q0 = total amount of
money spent on Q0 goods.
If Q < Q0  P > P0. SBCD =
The
benefit
to
the
consumer of paying the
fixed
value
of
P0
Consumer’s surplus (CS)
CS =
Q0
 f (Q )dQ − P0Q0
0
=
EXAMPLE
-------------------------------------------------------------------------------------------------------------------------------
Find the consumer’s surplus at Q0 = 5 for the demand
function P = 30 – 4Q
PRODUCER’S SURPLUS
------------------------------------------------------------------------------------------------------------------------------------
Supply function P = g(Q) &
Q = Q0  P = P0. SOABC =
P0Q0 = total amount of
money
received
on
Q0
goods. If Q < Q0  P < P0.
SBCD = The benefit to the
producer of selling at the
fixed
value
of
P0
Producer’s surplus (PS)
PS = P0Q 0 −
Q0
 g (Q )dQ
0
=
EXAMPLE
-------------------------------------------------------------------------------------------------------------------------------
Given the demand function P = 35 − Q 2 & supply functions
P = 3 + Q 2 , find the producer surplus at equilibriu m.