Applications of the exact integral. Plan: 1 Integral Definition 2

Applications of the exact integral.
Plan:

1 Integral Definition
2 Types of Integrals

There is a number of methods of calculations among which are functions, differentiation and integration. Application of Integrals is applied in various fields like Mathematics, Science, Engineering etc. For the calculation of areas, we use majorly integrals formulas. So let us give here a brief introduction on integrals based on the Mathematics subject to find areas under simple curves, areas bounded by a curve and a line and area between two curves, and also the application of integrals in the mathematical disciplines along with the solved problem.

Integral Definition
An integral is a function, of which a given function is the derivative. Integration is basically used to find the areas of the two-dimensional region and computing volumes of three-dimensional objects. Therefore, finding the integral of a function with respect to x means finding the area to the X-axis from the curve. The integral is also called as anti-derivative as it is the reverse process of differentiation.
Types of Integrals
There are basically two types of integrals, Definite and Indefinite. Definite Integral is defined as the integral which contains definite limits,i.e., upper limit and lower limit. It is also named as Riemann Integral. It is represented as;
We already examined exponential functions and logarithms in earlier chapters. However, we glossed over some key details in the previous discussions. For example, we did not study how to treat exponential functions with exponents that are irrational. The definition of the number e is another area where the previous development was somewhat incomplete. We now have the tools to deal with these concepts in a more mathematically rigorous way, and we do so in this section.



With the above example problem, we hope the concept of integrals is understood. In the same way, we can apply integrals to find the area of enclosed in eclipse, the area of the region bounded by the curve or for any enclosed area bounded in the x-axis and y-axis.
The application of integrations in real life is based upon the industry types, where this calculus is used. Like in the field of engineering, engineers use integrals to determine the shape of building constructions or length of power cable required to connect the two substations etc. In Science, it is used to solve many derivations of Physics topics like the centre of gravity etc. In the field of graphical representation, where three-dimensional models are demonstrated.
The application of integrals class 12 syllabus covers to find the area enclosed by the circle and similar kind of question pattern.
For more related topics of Integrals and NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives download BYJU’S- The Learning App.


In this chapter, we use definite integrals to calculate the force exerted on the dam when the reservoir is full and we examine how changing water levels affect that force. Hydrostatic force is only one of the many applications of definite integrals we explore in this chapter. From geometric applications such as surface area and volume, to physical applications such as mass and work, to growth and decay models, definite integrals are a powerful tool to help us understand and model the world around us.
Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. To find the area between two curves defined by functions, integrate the difference of the functions. If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals. In this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution. We can use this method on the same kinds of solids as the disk method or the washer method; however, with the disk and washer methods, we integrate along the coordinate axis parallel to the axis of revolution. With the method of cylindrical shells, we integrate along the coordinate axis perpendicular to the axis of revolution. The arc length of a curve can be calculated using a definite integral. The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of y. The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution. The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. In this section, we examine some physical applications of integration. Several physical applications of the definite integral are common in engineering and physics. Definite integrals can be used to determine the mass of an object if its density function is known. Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem. Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid.
One of the most prevalent applications of exponential functions involves growth and decay models. Exponential growth and decay show up in a host of natural applications. From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature. In this section, we examine exponential growth and decay in the context of some of these applications.
In this section, we consider centers of mass (also called centroids, under certain conditions) and moments. The basic idea of the center of mass is the notion of a balancing point. Many of us have seen performers who spin plates on the ends of sticks. The performers try to keep several of them spinning without allowing any of them to drop. Mathematically, that sweet spot is called the center of mass of the plate. We have looked at the definite integral as the signed area under a curve. This lets us compute total profit, or revenue, or cost, from the related marginal functions. We have looked at a number of applications where this was interpreted as an accumulation over time, including total production of an oil well and present value of a revenue stream. For some applications we want to look at the area between two curves. For example, considering profit as the area between the cost and revenue curves.
In this section we will look at more applications from finance and economics where the concepts can easily be described in terms as of the area between curves.
However, in that model, there were people who were willing to sell for less than the equilibrium price and people who were willing to buy for more than the equilibrium price. These people got an exceptionally good deal in the transaction. We would like to measure that benefit, since we can think of it as the extra profit the suppliers and buyers make on the transaction. We note that each side will have an incentive to maximize that benefit.
Focus first on the consumer side. The area under the demand function, from 0 to the quantity sold, measures the consumers’ willingness to spend. The area in the rectangle with that same base and height equal to the sale price measures the actual consumer expenditure. The difference between the two is a quantity we will call