What is the purpose of integral calculus and how it is applied in calculus?

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In mathematics, Integral calculus or integration is one of the two types of calculus. Another type of calculus is differential calculus. In mathematics, Calculus which is also known as infinitesimal calculus is a branch of mathematics that studies continuous change in the unique way that algebra studies arithmetic operations and geometry studies shape in general.

To improve the design of important infrastructure we use calculus, such as bridges and buildings. We can also use calculus in Electrical Engineering, we use integral calculus to calculate the exact length of the electrical line required to connect two substations that are thousands of miles apart.

What is Integral Calculus?

Integral calculus, like differential calculus, is an important component of calculus. We explore the relationship between two quantities in differential calculus, such as distance and time.

However, in integral calculus, the inverse process of a relationship between two quantities is used. Integration, anti-differentiation, or anti-derivative are terms used to describe this process.

Integral calculus can be stated as the study of the properties, definitions, and applications of two related notions, the definite and indefinite integrals. Integration is the procedure for calculating the value of an integral.

Integral calculus is a type of calculus of mathematics that studies two linear operators that are related. Integral calculus is a type of calculus that is used for calculations involving arc length, pressure, area, center of mass, volume, and work. Calculus can also help you comprehend the nature of space, time, and motion more precisely.

Types of Integral Calculus

Definite integral

An integral which takes a function as input and returns a number that signifies the algebraic sum of areas among the input graph and the x-axis is said to be the definite integral. The limit of a Riemann sum of rectangular areas is used in the technical definition of the definite integral.

We can represent this type of integral as,

 = F(b) – F(a)

Here, ʃ this notation is called integration symbol, a is the lower limit of the function, b is the upper limit of the function, and dx is the integrating agent.

Some basic properties of this type of integral are given below.

  • f(x) dx = f(u) du
  • f(x) dx = – f(x) dx
  • f(x) dx = 0
  • f(x) dx = f(x) dx + f(x) dx
  • f(x) dx = f (a – x + b) dx
  • f(x) dx = f(a – x) dx

In calculus, we can use definite integrals to find the area under, over, and between curves. The area between the curve and the x-axis equals the definite integral of the function if the given function is strictly positive,  in the specified interval. The area of a negative function is equal to -1 times the definite integral. Definite integrals can also be referred to as antiderivative.

All the problems of integral can easily be calculated by using an antiderivative calculator.

Indefinite integral

In calculus, the indefinite integral is the main type of integral which is stated as the inverse operation of the differential integral. When f is a derivative of F, F is an indefinite integral of f. Calculus makes frequent use of lower- and upper-case letters for functions and their indefinite integrals.

We can represent this type of integral as,

ʃf(x) dx = F(x) + C

where R.H.S. is basically for integral of f(x) with respect to x in the given equation.

F(x) is known as an anti-derivative or primitive function.

The integrand is defined as f(x).

The integrating agent is known as dx.

The integration constant is denoted by the letter C.

And the integration variable is x.

Some basic formulas of this type of integral are given below.

  • ∫1 dx = x + C
  • ∫ O dx = Ox + C
  • ∫ xndx = ((xn+1)/(n+1)) + C; n ≠ 1
  • ∫cosec(2x) dx = -cot(x) + C
  • ∫cos(x) dx = sin(x) + C
  • ∫sin(x) dx = – cos(x) + C
  • ∫sec(x) * tan(x) dx = sec(x) + C
  • ∫sec(2x) dx = tan(x) + C
  • ∫cosec(x) * cot(x) dx = -cosec (x) + C
  • ∫(1/x) dx = ln|x| + C
  • ∫ exdx = ex + C
  • ∫ axdx = (ax/ln a) + C; a > 0, a ≠ 1

To calculate the perfect output of your problems related to integral, you can use the integral calculator.

It can be represented graphically as an integral symbol, a function, and finally a dx. The indefinite integral is a more straightforward way of expressing the antiderivative. Even the definite integral and the indefinite integral are comparable, they are not the same. The distinction between definite and indefinite integrals is illustrated in the diagram below.

Applications of Integral calculus

The area of the region circumscribed by the curve, any enclosed area bounded in the x-axis and y-axis, or the area enclosed in the eclipse all of these can be found using integrals.

Integral calculus is widely used in mathematics and physics and has a wide application in these subjects. Some of them are mentioned below.

Application of integral calculus in mathematics.

  • A region with curved sides has a center of mass (Centroid).
  • A curve’s average value.
  • The space that exists between two curves
  • The area enclosed by a curve

Application of integral calculus in Physics.

  • Gravitational center
  • The mass centers
  • Vehicle mass and moment of inertia
  • Satellite mass and momentum
  • A satellite’s velocity and trajectory
  • Thrust

Summary

Integral calculus is a type of calculus widely used in mathematics. Integral calculus can be stated as the study of the properties, definitions, and applications of two related notions, the definite and indefinite integrals.

Integration is the procedure for calculating the value of an integral. Integral has two types one is the indefinite integral and the other type of integral is a definite integral. An integral in which limits are involved is said to be the definite integral while the integral which is not definite is indefinite integral.

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