Differential calculus measures the rate of change of functions.

**Integral
Calculus**

Differential
calculus measures the rate of change of functions. In Economics it is also
necessary to reverse the process of differentiation and find the function F(x)
whose rate of change has been given. This is called integration. The function
F(x) is termed an integral or anti- derivative of the function f(x).

The
integral of a function f(x) is expressed mathematically as

Here the
left hand side of the equation is read “the integral of f(x) with respect to x”
The symbol ∫ is an integral sign, f(x) is integrand, C is the constant of
integration, and F(x)+c is an
indefinite integral. It is so called because, as a function of x, which is here
unspecified, it can assume many values.

If the
differential coefficient of *F* (x)
with respect to x is f(x), then an integral of f(x)

with
respect to x is F(x) . It is a reverse process of differentiation. In symbols:

Following
points need to be remembered:

a. ∫ is used to denote the process of integration.
In fact, this symbol is an elongated ‘S’ denoting sum.

b. The
differential symbol ‘dx’ is written by the side of the function to be
integrated.

c. ∫ f x dx = F(x)+C, C is the integral constant

∫ f x dx means, integration of f(x) with
respect to x.

__Example 12.26__

Let the
marginal cost function of a firm be 100-10x+0.1x^{2} where x is the output. Obtain the total cost function
of the firm under the assumption that its fixed cost is ** Rs.**500.

__Example 12.27__

The
marginal cost function for producing x units is y = 23 + 16x - 3x^{2} and the total cost for producing zero unit is ** Rs.**40.
Obtain the total cost function and the average cost function.

__Solution :__

Given the
marginal cost function y = 23 + 16x - 3x2 ; C = 40

** Rs.**40 is the
fixed cost.

We know
that

Total
Cost function = ∫ (Marginal cost function) dx+C

This
theory was developed by the Alfred Marshall. The demand function P(x) reveals
the relationship between the quantities that the people would buy at given
price. It can be expressed as

P = *f* (x)

Consumer
surplus is the difference between the price one is willing to pay and the price
that is actually paid.

It is
represented in the following diagram.

Mathematically,
the consumer’s surplus (CS) can be defined as

__Example:12.28__

If the
demand function is P = 35 - 2x - x^{2} and the demand x_{0}
is 3, what will be the consumer’s surplus?

__Solution__

Given
demand function,

P = 35 - 2x - x^{2}

for x = 3

= 35 - 2(3) -3^{2}

=35 - 6 - 9

P = 20

Therefore,

CS = (Area of the curve below the demand curve from 0 to 3) - Area of the rectangle (20 x 3 = 60)

__Example 12.29__

Given the
demand function P_{d} = 25 - Q^{2} and the supply function P_{s} = 2Q + 1.

Assuming
pure competition, find (a) consumers surplus and (b) producers surplus. (P_{d}
=Demand price; P_{s} = Supply
price)

__Solution:__

For
market equilibrium, *P _{d}*

25-Q^{2} = 2Q+1

= -25 + Q^{2} + 2Q+1

0 = -24 + Q^{2} + 2Q

Q^{2}
+ 2Q - 24 = 0

Q2 + 6Q - 4Q - 24 = 0

Q(Q + 6) -4(Q + 6) = 0

(Q + 6)(Q- 4) = 0

So, Q = 4 or Q = -6. Since
Q cannot be equal to -6,

Q = 4

=36 - (16 + 4) = 16

**Think
and Do**

Find your change in mark by additional hour of study in any of
your subject

Find your consumption of petrol for an additional unit of
kilometer travelled

Ask your parents about their spending with respect to every
additional unit of wage or salary or income

Tags : Integration, Meaning, Basic Rule, Formula, Solved Example Problems, Application | Economics , 11th Economics : Chapter 12 : Mathematical Methods for Economics

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11th Economics : Chapter 12 : Mathematical Methods for Economics : Integral Calculus | Integration, Meaning, Basic Rule, Formula, Solved Example Problems, Application | Economics

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