## Numerical integration

** Calculation of a definite integral. ** There are various methods of numerical integration. In particular, to calculate the approximate value of a definite integral using the trapezoid method, one can use the formula

where the segment [* a; b *] is divided into

*equal intervals,*

**n***0 =*

**x***n =*

**a, x**

**b.**The method of calculating the approximate value of a definite integral in MS Excel is based on computing the numerical values of the integrand for each integration step.

** Example 9.19. ** It is required to calculate the definite integral in the interval from 1 to 3 in increments of 0.2 (obviously, the smaller the step , the more accurate the result).

* Solution *

Let's do the following:

• We will create a region for the argument of the integrand function on the interval from 1 to 3 in increments of 0.2.

• We compute the values of the integrand for each argument.

• To the right of the values of the function in cell C2, enter the formula = (A3-A2) * B2 + (A3-A2) * (B3-B2)/2, which calculates the area of the first part of the trapezoid.

• Copy the formula to the value of the argument 2.8.

• In the cell below, we calculate the sum of the values of the column created above. The calculated value is the value of a certain integral (Figure 9.36).

* Fig. 9.36. *

**Approximate calculation of a definite integral**

** Calculation of limit values. ** The cost of products primarily depends on the volume produced: * S = F * (

*) The marginal cost is determined by the cost (*

**Q***) of the product growth (Δ*

**ΔS****Q**) and is expressed by the formula . In the case where the dependence of is continuous, Δ instead of the above particular equation, we can use the derivative: border = 0 src="images/image143.jpg">.

** Example 9.20. ** Consider the problem in which the dependence of output on production costs in terms of money is characterized by the formula * S * =

*< strong> IQ -*0.005Q3. It is necessary to determine the marginal costs of production, if the output is 10 den. units

* Solution *

We introduce into the cells the neighborhood of point 10 (the left and right approximations).

Let's calculate the values of the function that characterizes the dependence of the output volume on production costs, at these points.

Using the formula for finding the derivative, we define the marginal cost of production (Figure 9.37).

* Fig. 9.37. *

**Marginal production costs**

** Calculation of the elasticity of economic indicators. ** The concept of elasticity of demand in economic theory is used in the analysis and forecasts of pricing policy. The elasticity of demand characterizes the dependence of the percentage change in demand when the price of a commodity changes by one percent and is expressed by the formula

where * D * (

*) is the demand function for the product;*

**p***is the price of the product.*

**p**Determine the elasticity of demand, consider the following example.

** Example 9.21. ** The demand for a product is determined by the formula * D * (

*). = 1000 - 30*

**p***. Determine the elasticity of demand at a price of 20 den. units*

**p** * Solution *

Define the marginal cost of production at a price * p = * 20.

Apply the formula of elasticity of demand at a price of 20 den. units (Figure 9.38).

* Fig. 9.38. *

**Calculation of the elasticity of demand**

** Calculation of integral indicators. ** Let's consider the problem of this type by the example of determining the cost of cargo transportation by rail.

** Example 9.22. ** The cost of transporting one ton of cargo is reduced by 500 rubles. with every kilometer. The initial freight transportation rate for the first kilometer is 15,000 thousand rubles. Determine the cost of transporting 20 tons of cargo for a distance of 20 km.

* Solution *

Create an area in MS Excel for the range from 0 to 20 (distance in kilometers).

For each kilometer, calculate the cost of shipping the goods.

We apply the formula for calculating a definite integral for each kilometer from 0 to 19, and in the line corresponding to the 20th kilometer, calculate the sum of the obtained column, which is the total cost of transporting 20 tons of cargo by rail (Figure 9.39).

* Fig. 9.39. *

**Shipping by rail**

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