IntroductionAs the students of M.B.A, we need to know about many things regarding business mathematics. An entrepreneur needs to take many decisions with the help of mathematical terms. By using the terms properly, an entrepreneur can easily take any decision quantitatively. Among many tools, Integral calculus is one of them. It helps to identify many things related to our practical business life. Such as: total cost, total revenue, producer’s surplus, consumer’s surplus etc. So, it’s an important element of business tools. This report is completely based on integral calculus and it’s uses in different fields.

ObjectivesThe main objectives of this report are:

1. To know about integration in details.

2. To know the methods and techniques of integration.

3. To know the uses of integral calculus in our day to day life.

IntegrationIntegration from the Latin integer meaning whole or entire generally means combining parts so that they work together or form a whole. Integration is often introduced as the reverse process to differentiation, and has wide applications, for example in finding areas under curves.

Process of Integration:

Types of Integration: There are basic two types of integration. They are:

ECTIVES

Definite Integral:

Techniques of Integration: There are some major techniques of integration are:

Integration as summation

Integration may be introduced as a means of finding areas using summation and limits

Integration using a table of anti-derivatives

Integration may be regarded as the reverse of differentiation, so a table of derivatives can be read backwards as a table of anti-derivatives.

Integration by parts

A special rule, integration by parts, can often be used to integrate the product of two functions. It is appropriate when one of the functions forming the product is recognised as the derivative of another function. The result still involves an integral, but in many cases the new integral will be simpler than the original one.

Integration by substitution

There are occasions when it is possible to perform an apparently difficult piece of integration by first making a substitution. This has the effect of changing the variable and the integrand. With definite integrals the limits of integration can also change.

Application Of Integration:

· define Total Cost, Variable Cost, Average Cost, Marginal Cost, Total Revenue, Marginal Revenue and Average Revenue;

· find marginal cost and average cost when total cost is given;

· find marginal revenue and average revenue when total revenue is given;

· find total cost/ total revenue when marginal cost/marginal revenue are given, under given conditions.

. define and calculate a consumer’s surplus

. define and calculate a producer’s surplus.

Determination of cost function:

If C denotes the total cost and MC is the marginal cost, then we can writeC Cx MCdx k , where k is the constant of integration, k, being the constant, is the fixed cost.

Example: Given MC = 5+16x-3x2

C(x) = (5+16x-3x2)dx

C(x) = 5x +8x2 -x3 +k

When x = 5, C(x) = C(5) = Rs. 500

or, 500= 25+200-125+ k

This gives, k = 400

C(x)= 5x +8x2 -x3 +400

Determination of Total Revenue Function:

If R(x) denotes the total revenue function and MR is the marginal revenue function, then

R(x)= (MR)dx+k Where k is the constant of integration. R(X)

Also, when R (x) is known, the demand function can be found as p= X

Example:

The marginal revenue function of a commodity is given as

MR =12-3x2 +4x . Find the total revenue function.

MR = 12-3x2 +4x

R = (12-3x2 +4x)dx+k

R =12x-x3 +2x2 [constant of integration is zero in this case]

Revenue function is given by R = 12x +2x2-x3

Average Value of a Function Over an Interval:

EXAMPLE:

During a certain 12-hour period the temperature at time t (measured in hours from the start of the period) was degrees. What was the average temperature during that period?

SOLUTION:

The average temperature during the 12-hour period from t = 0 to t = 12 is

Consumer surplus:

The consumer surplus represents the total savings to consumers who are willing to pay more than price for the product.

The consumer’s surplus at a price level of p is

Where is demand function and pq is the revenue.

EXAMPLE:

Find the consumers’ surplus for the following demand curve at the given

sales level x.

SOLUTION:

Therefore, the consumers’ surplus is

That is, the consumers’ surplus is $20. Graphical Presentation of Consumers’ Surplus.

• On the graph, the consumer surplus (yellow) is the area located below the Demand function and above the rectangle that represents the revenue generated (red).

Supplier’s/ Producer’s surplus: The supplier’s surplus measures the difference between the amount of money a supplier is willing to accept at a given price for a product and the amount the supplier actually does receive.

The supplier’s surplus at a price level of p is

Where is supply function and pq is the revenue.

EXAMPLE:

At market equilibrium , consumers demand 100 (000) tons if SAE 90 lubricating oil, whose supply function is Ps (q) = 10+0.5 q

Where q is the thousands tons and ps(q) is in dollars per ton. Compute producer’s surplus.

SOLUTION:

Since demand is 100 thousand tons, so

P(100) = 10+0.5(100)

= 10+50= 60 (thousands)

Therefore, producer’s surplus is

= 6000- [10q+0.5q2/2] 0100

= 6000-(1000+0.5X100X100/2)-0-0

=6000-(1000+2500)

= 6000-3500=2500 thousands

Graphical Presentation Of Producer / Suppliers surplus:

The area (yellow) above the Supply function and still in the rectangle representing income is the producer surplus.

Graphical Presentation Of Consumers and Producer / Suppliers surplus

Conclusion

From the above discussion we can say that, integral calculus is an important part of business mathematics. We can use it in many ways and take effective decisions. However it easily determines different terms those can not be done easily in other ways. If anybody knows its function and properties he/she can apply it to determine many important things easily.