Assistant Professor, Department of mathematics Kalinga University, Naya Raipur, Chhattisgarh
People who like mathematics need not explain much what the importance of mathematics is in real life. But there are more people who start having fever when they hear the word “mathematics”, and somewhere such people must have started feeling that wow! that is the mathematics. It is because of this mathematics that he keeps track of his routine. How much did I earn today? How much did I spend today? If I earn so much money every day, I will earn a lot of money in a few days and so on.
The interpretation of the area and volume of two dimensional and three dimensional regular geometric shapes with regular structure is based on geometric and algebraic subjects, but the area of a geometrical shape that is irregular is not easy to calculate by geometric methods only. Also, it is not possible to find the volume of solids formed by rotating a binary shape only through the concepts of geometric mathematics. The help of a major branch of mathematics, which we call calculus, is taken.
There are some applications given which are used in our life.
Using the concept of integral calculus above type of problem can be solved and calculated the area approximately. There are some methods which are taken from the calculus called as numerical methods. Let us see one of the methods “Simpsons 1/3rd rule”. Let us find the area a n irregular shape given as below
To find the area of this irregular shape we take
Here the irregular shape is taken on the graph, each cell is of 1×1 square cm.
The horizontal line AB is in the shape, this line is equally divided up into 9 parts by taking d=2cm, here the heights in the shape are taken as 𝑦𝑖, where i= 0, 1, 2,…,10.It is clear that 𝑦0 and 𝑦10 both are having 0 and 𝑦1 =3 cm, 𝑦2 =4 cm, 𝑦3 = 4cm, 𝑦4 = 5cm, 𝑦5 = 6cm, 𝑦6 =7 cm, 𝑦7 = 5cm, 𝑦8 =4 cm, 𝑦9 =2 cm.
Now using the Simpson’s 1/3rd rule
∫ 𝑦𝑑𝑥 = 𝑑 3 [(𝑦0 + 𝑦𝑛) + 4(𝑦1 + 𝑦3 + 𝑦5 + ⋯ + 𝑦𝑛−1) + 2(𝑦2 + 𝑦4 + 𝑦6 + ⋯ + 𝑦𝑛−2)]
Here d= 2cm,
∫ 𝑦𝑑𝑥 = 2/3[(0 + 0) + 4(3 + 4 + 6 + 5 + 2) + 2(4 + 5 + 7 + 4)]
=2/3 [ 0+4×20 +2×20]
=2/3 [80+40]
=80sqcm
Hence the area of given irregular shape is 80sqr cm approximately obtained
Application in the economic
In the economic zone, to calculate the optimum profit the concept of second differential coefficient is applied. In the economic and commercial filed three types of functions are discussed named as cost function to produce the materials, The value of the production function, revenue function and profitable function.
Total cost function (C(x) = It is the complete economic cost of production and it is with variable
cost. If x be the level of output, then total cost function is represented as C(x)
Marginal cost M(x): It is the derivative of total cost function i.e., M(x) = 𝑑/𝑑𝑥 𝐶(𝑥).
Application in Engineering and Technology:
There are many engineering subjects that are beyond mathematics, calculus is the basis of engineering and technology. It is not possible, without knowledge of calculus, to calculate the rate of change with respect to time of a moving parts like machine parts or automobile.
There are some applications in the field of engineering and technology:
References:
1. https://owlcation.com/stem/Calculating-the-Area-of-Irregular-Shapes-Using-Simpsons13-Rule-Area-Approximation.
2. Marsitin, R. (2019, November). Analysis of Differential Calculus in Economics. In Journal of
Physics: Conference Series (Vol. 1381, No. 1, p. 012003). IOP Publishing.
3. https://sciencing.com/calculate-hydraulic-flow-7155002.html.
4. Papst, I. (2011). A biological application of the calculus of variations. The Waterloo Mathematics Review, 1, 3-16
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