An importance of Calculus in real world

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).

1. Find the first derivative of C(x) and equating it equal to zero, we can get the level of output x, i.e find 𝑑/𝑑𝑥 𝐶(𝑥) and put 𝑑/𝑑𝑥 𝐶(𝑥) =0 to get value of x.
2. Find second derivative of C(x) i.e. 𝑑2 2𝐶(𝑥)/𝑑𝑥2 2 at the level of output obtained in (I).
3. If 𝑑2𝐶(𝑥) 𝑑𝑥2 > 0 at the level of output x = c, the total cost is minimum.

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:

• To derive the fundamental as well as advance equations used in mechanics.
• To calculate the bearing capacity and shear strength.
• To obtained the area of the surface like complex objects.
• Rocket equation can be derived and solved the differential equations, and many more