Integration: From Marginal Utility to Definite Numeric Solutions

Integration (Indefinite Integration):
  
If the Marginal Utility Function is 2q, then the Total
Utility Function is q2.
  
That is, if derivative is given, our problem is to find out
the function.
  
If the Differential co-efficient of F(x) with respect
to x is f(x), then an Integral of f(x) with respect to x
is F(x).
Notation for Indefinite Integration:
  
If d/dx f(x) = f(x), then the usual notation for the
indefinite integral is
    
∫f(x) dx
    
   or
    
 ∫ y dx = F(x) + C
Rule: I
  
ʃ dx= x+c
Rule: II
  
ʃK dx= K ʃdx=Kx+c
Rule: III
  
 ʃxⁿdx= 
×ⁿ+¹/ ⁿ+¹  
+C , n≠ -1
Rule: IV
 
  
Integral of sum or difference
   
A number of functions is equal to the sum or difference of
the Integrals of the separate function.
   
ʃ(dx
,+ dx+.....+dx
ñ 
) = ʃ dx
,+ 
ʃ
 dx+.....+  
ʃ 
dx
ñ
 
Rule: V
  
Integral of Multiple by a Constant
   
A function is multiplied by a constant number, this number
will remain a multiple of the integral of the function.
Example:
   
 ʃ4x²dx = 4 ʃx²dx
    
= 4 x²+¹/2+1 + C
    
= 4x³/3
Rule: VI
 
 Integration by Substitution
  
A product or quotient of two differentiable functions of x, it may be
possible to express them as a constant multiple of another function f(u) and
its derivative is du/dx,
   
ʃ f(x) dx= ʃ f(u).du
  
 
  
Rule: VII
 Logarithmic Function Rule:
  
  
y= log X
   
 ʃ 1/x dx= logx + C.
   
where C is constant.
Rule: VIII
  
Exponential Function Rule
  
y= ₑx
 
   
y= ʃ ₑx  dx=  ₑx +c.
  
  
In definite integration a definite numerical solution to the problem
is possible. In definite integration a limit is added in the process of
integration. There should be the lower and upper limits.
  
This is represented as .
   
This integral has upper limit as b and lower limit as a. If
f(x) is a continuous function on closed interval (a,b) and F(x) is an
integral function of f(x) then
    
   
 
    
-is called definite integration
 
         
     
 
 
 
Notation for Definite Integration
   
It is to be read as the definite integral of y from
x=a to x=b.
    
 
I) If we interchange the limiting values of integration,
only the sign of the definite integral changes; thus,
II) The limits can be broken. A definite integral can be
expressed as a sum of a finite number of definite sub-
integrals as follows.
 
 
Where c
1 
 and c
2
 are values in between a and b.
III ) The value of definite integral is independent of the
variable of the integration
IV)
V)   Substitution necessitates changes in the limits
 
 
   
Applications in Economics
  
indefinite integration is used in economics to derive the total
functions from the marginal functions. i.e.
Total cost function
    
It is an integral of marginal cost function C=∫
MC dx.
Total Revenue Function
    
It is an integral of marginal revenue function
R= ∫ MR dx.
Total Utility Function
 
   
    
 It is an integral of marginal utility function
U= ∫MU dx.
   
   
It is a function of income i.e. c=f(y). It is a
integral function of MPC
    
 C= ∫ MPC dy
Savings Functions. S= f(y)
    
It is the integral function of MPS.
    
S= ∫ MPS dy.
Cost Function:
   
MC is given, we have to find out the total
cost(c)
   
i.e. C= ∫ MC.dx
    
= ∫ dc/dx. dx
    
Revenue Functions:
   
   
MR is given, we have to find out the Total
revenue (R or TR)
   
R= ∫ MR.dx
    
= ∫ dR/dx.dx
Applications of Definite Integration
   
   
Capital Formation
   
Area between Two curves
   
Consumer’s Surplus
   
Producer’s Surplus
 
 Capital formation means increase in the real stock of
capital during a given period.
 
Rate of  capital Formation may thus be expressed as
    
dk/dt
 
K(t)= ∫ I(t) dt= ∫ dK/dt .dt =∫ dK
  
Area =
   
    
Consumer’s Surplus is the difference between the price
that a consumer is willing to pay for a commodity
rather than go without it and the actual price he pays
for the commodity
 
 
Consumer’s surplus= Potential price- Actual price
 
1.
Given the demand function p= 35-2x-x², find
consumer’s surplus at x=3.
2.
 If the demand function at p= 85-4x-x², find
consumer’s surplus at p
o
= 64
3.
 The demand function for a commodity p=20-3x. The
supply function p=2x. Find consumer’s surplus.
   
   
   
P.S= Poxo -
 
1.
The supply function for a commodity p=2+D². Find
the Producer’s surplus when p=18.
2.
Find the producer’s surplus for the supply function
p=2x²+x+5. when x
0
=3.
3.
The demand function for a commodity p=25D-20. the
supply function p=5D+60
Consumer’s surplus and Producer’s surplus:
1.
 Given the demand function p=8-2x and the supply
function p=2+x, find the consumer’s surplus and the
producer’s surplus.
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Integration plays a crucial role in mathematics, connecting concepts such as marginal utility and total utility functions, differential coefficients, rules for indefinite integration, and the process of definite integration. Learn about notation, rules, and examples to grasp the fundamentals and applications of integration.


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  1. Integration (Indefinite Integration): If the Marginal Utility Function is 2q, then the Total Utility Function is q2. That is, if derivative is given, our problem is to find out the function.

  2. If the Differential co-efficient of F(x) with respect to x is f(x), then an Integral of f(x) with respect to x is F(x). Notation for Indefinite Integration: If d/dx f(x) = f(x), then the usual notation for the indefinite integral is f(x) dx or y dx = F(x) + C

  3. Rule: I dx= x+c Rule: II K dx= K dx=Kx+c Rule: III x dx= + / + +C , n -1 Rule: IV Integral of sum or difference A number of functions is equal to the sum or difference of the Integrals of the separate function. (dx,+ dx+.....+dx ) = dx,+ dx+.....+ dx

  4. Rule: V Integral of Multiple by a Constant A function is multiplied by a constant number, this number will remain a multiple of the integral of the function. Example: 4x dx = 4 x dx = 4 x + /2+1 + C = 4x /3 Rule: VI Integration by Substitution A product or quotient of two differentiable functions of x, it may be possible to express them as a constant multiple of another function f(u) and its derivative is du/dx, f(x) dx= f(u).du

  5. Rule: VII Logarithmic Function Rule: y= log X 1/x dx= logx + C. where C is constant. Rule: VIII Exponential Function Rule y= x y= x dx= x +c.

  6. In definite integration a definite numerical solution to the problem is possible. In definite integration a limit is added in the process of integration. There should be the lower and upper limits. This is represented as . This integral has upper limit as b and lower limit as a. If f(x) is a continuous function on closed interval (a,b) and F(x) is an integral function of f(x) then -is called definite integration

  7. Notation for Definite Integration It is to be read as the definite integral of y from x=a to x=b.

  8. I) If we interchange the limiting values of integration, only the sign of the definite integral changes; thus, II) The limits can be broken. A definite integral can be expressed as a sum of a finite number of definite sub- integrals as follows.

  9. Where c1 and c2 are values in between a and b. III ) The value of definite integral is independent of the variable of the integration IV) V) Substitution necessitates changes in the limits

  10. Applications in Economics indefinite integration is used in economics to derive the total functions from the marginal functions. i.e. Total cost function It is an integral of marginal cost function C= MC dx. Total Revenue Function It is an integral of marginal revenue function R= MR dx. Total Utility Function It is an integral of marginal utility function U= MU dx.

  11. It is a function of income i.e. c=f(y). It is a integral function of MPC C= MPC dy Savings Functions. S= f(y) It is the integral function of MPS. S= MPS dy.

  12. Cost Function: MC is given, we have to find out the total cost(c) i.e. C= MC.dx = dc/dx. dx

  13. Revenue Functions: MR is given, we have to find out the Total revenue (R or TR) R= MR.dx = dR/dx.dx

  14. Applications of Definite Integration Capital Formation Area between Two curves Consumer s Surplus Producer s Surplus

  15. Capital formation means increase in the real stock of capital during a given period. Rate of capital Formation may thus be expressed as dk/dt K(t)= I(t) dt= dK/dt .dt = dK

  16. Area =

  17. Consumers Surplus is the difference between the price that a consumer is willing to pay for a commodity rather than go without it and the actual price he pays for the commodity Consumer s surplus= Potential price- Actual price

  18. Given the demand function p= 35-2x-x, find consumer s surplus at x=3. If the demand function at p= 85-4x-x , find consumer s surplus at po= 64 The demand function for a commodity p=20-3x. The supply function p=2x. Find consumer s surplus. 1. 2. 3.

  19. P.S= Poxo -

  20. The supply function for a commodity p=2+D. Find the Producer s surplus when p=18. Find the producer s surplus for the supply function p=2x +x+5. when x0=3. The demand function for a commodity p=25D-20. the supply function p=5D+60 1. 2. 3.

  21. Consumers surplus and Producers surplus: Given the demand function p=8-2x and the supply function p=2+x, find the consumer s surplus and the producer s surplus. 1.

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