Integral Calculus: Definition, Theorem, Formula, Application

Integral Calculus: Integral calculus is the branch of calculus where we learn about the theory, properties, and applications of integral. It is closely related to differential calculus and together leads to the foundation of mathematical analysis. The integral calculus and differential calculus are connected with the fundamental theorem of calculus.

Integral calculus deals with lengths, areas, volumes, total values, and the derivation of the formula for finding antiderivatives. Let’s dive in to know more about it.

What is Integral Calculus?

Calculus (singular) or Calculi (plural) is a Latin term that means “small pebble” – small stones used by the Romans to count in their abacus. It was used to count inexhaustibly small numbers. Calculus is a type of mathematics that is constantly in motion. It’s a method of calculating change that’s used to explain our world’s complex existence.

The analysis of the definitions, properties, and applications of two similar terms, the indefinite integral and the definite integral, is known as integral calculus. Integration is the method of determining the value of an integral. Integral calculus is a branch of mathematics that studies two connected linear operators. Integration is a crucial term since it is the inverse process of differentiation.

Fundamental Theorem of Integral Calculus

The fundamental theorem of calculus relates the concepts of differentiating and integrating functions together. The first part of the theorem – the first fundamental theorem of calculus states that one of the antiderivatives (also known as an indefinite integral) of a function \(f,\) say \(F,\) can be obtained by integrating \(f\) with a variable bound of integration. For continuous functions, this implies the presence of antiderivatives.

The second part of the theorem – the second fundamental theorem of calculus states that the integral of a function \(f\) over some interval can be computed using any of its infinitely many antiderivatives, such as \(F.\) There is an infinite number since there is an infinite number of options for the integration constant, \(C.\)

This section of the theorem is useful in practice since it avoids using numerical integration to calculate integrals by specifically finding the antiderivative of a function using symbolic integration. This usually results in a higher level of numerical precision.

The Fundamental Theorem of integral calculus connects the derivative and the integral, and it’s the most common way to evaluate definite integrals. In a nutshell, it states that every continuous function over an interval has an antiderivative (a function whose rate of change, or derivative, equals the function). Furthermore, the difference \(F(b)-F(a),\) where \(F\) is the function’s antiderivative, is the definite integral of such a function over an interval \(a<x<b .\)

Categories of Integral Calculus

Integral calculus is divided into two categories. They are:

1. Indefinite Integral
2. Definite Integral

Indefinite Integral

A function that takes the antiderivative of another function is called an indefinite integral. The indefinite integral is not defined with the help of upper and lower limits. Indefinite integral represents the family of the function whose derivatives are \(f.\) The difference between any two functions within the family is a constant.

By simply omitting the integration limits, the integral key, which is used to find definite integrals, can also be used to find indefinite integrals.If the integration of \(f(x)\) is \(F(x).\) Then it is represented by:

\(\int f(x) d x=F(x)+C\)
which is read as Integral of \(f(x)\) with respect to \(x\) and equals to \(F(x)\) plus \(C,\) where,
\(F(x)\) is Antiderivative or Primitive
\(f(x)\) is the Integrand
\(dx\) is the Integrating agent 
\(x\) is the variable of Integration
\(C\) is Constant of Integration

So, there exist an infinite number of antiderivatives of a function \(f.\)
For example, if \(F(x)=x^{2}+5,\) and \(F(x)=x^{2}+1,\) 

Since the derivative of both functions are the same i.e. \(f(x) =2x\)
So, if we change the constant term of \(F(x)\) to another constant term, every time we get its derivatives to be the same.

Definite Integral

A definite integral has upper and lower limits (i.e., a start and an end value). \(x\) is limited to lying on a real line. When restricted to lie on the real line, a definite integral is also known as a Riemann Integral.
The following is a representation of a definite integral:

\(\int_{a}^{b} f(x) d x\), where
\(a\) and \(b\) are lower and upper limits respectively. \(f\left( x \right)\) is called the integrand and \(dx\) is called the integrating agent.
One of the applications of the definite integral is to calculate the area bounded by curves with coordinate axes.

What is Integral Calculus Used for?

Calculus is a branch of Mathematics that determines how matter, particles, and celestial bodies move. The rate of change in real-time is calculated using calculus. Isaac Newton applied calculus directly to measuring physical structures.

Mathematicians could use these methods to measure slopes, curvatures, rate of change, and motion. It allowed them to comprehend motion and dynamics in an evolving universe, as well as the elements in the earth and galactic planet orbits.

Integral calculus is primarily used for two purposes. One is to find \(f\) from \(f’\) and the other is to find out the area under the curve.

A general function \(f(x)\) is sometimes positive and sometimes negative, so the integral calculates the signed area, that is, the total area above the \(x\)-axis minus the total area below the \(x\)-axis. The integral count areas above/below the \(x\)-axis as positive/negative.

Integral Calculus Formulas

We have got some Integral formula which is generally used while calculating integral. Some of the important integral calculus formulas are given below:

(i) Constant Rule: \(\int k d x=k x+C\)

(ii) Constant Multiple Rule: \(\int k f(x) d x=k \int f(x) d x,\) where \(k\) is constant.

(iii) Sum/Difference Rule: \(\int f(x) \pm g(x) d x=\int f(x) d x \pm \int g(x) d x\)

(iv) Power Rule: \(\int x^{n} d x=\frac{x^{n+1}}{n+1}+C, n \neq-1\)

(v) Log Rule: \(\int{\frac{1}{x}dx}=\ln |x|+C,x\ne 0\)

(vi) Exponent Rule: \(\int a^{k x} d x=\frac{a^{k x}}{k \ln a}+C, x \neq 0\)

(vii) Trigonometric Rule:
\(\int \sin x d x=-\cos x+C\)
\(\int \cos x d x=\sin x+C\)
\(\int \tan x d x= \ln |\sec x|+C\) or \(-ln |\cos x|+C\)
\(\int {\cot xdx = \,\ln \,\left| {\sin x} \right| + C}\)
\(\int \sec x d x= \ln |\sec x+\tan x|+C\)
\(\int\mathrm{cosec}\;xdx= \ln |\operatorname{cosec} x-\cot x|+C\)
\(\int\sec^2xdx=\tan x+C\)
\(\int \sec x \tan x d x=\sec x+C\)
\(\int\mathrm{cosec}^2xdx=-\cot x+C\)
\(\int\tan^2xdx=\tan x-x+C\)

For Definite Integral
\(\int_{a}^{b} f(x) d x=F(b)-F(a)\)

Where,
\(F(x)\) is Antiderivative or Primitive
\(F(b)\) is value of Integral at Upper limit
\(F(a)\) is value of Integral at Lower Limit
\(f(x)\) is the Integrand
\(dx\) is the Integrating agent 
\(x\) is the variable of Integration

What are the Applications of Integral Calculus?

Calculus is used in actuarial science, computer science, statistics, engineering, economics, business, medicine, and demography, among other fields.

Calculus is used in many disciplines, including physics, chemistry, medicine, economics, biology, engineering, space exploration, statistics, and pharmacology. Architects and engineers can not construct stable structures without calculus. One can find the application of integral calculus in the area under curves, arc length, surface area, volume, probability, etc.

Solved Examples

Q.1. If  \(f'(x)=2x\) and \(f(0)=2,\) then find \(f(x).\)
Ans: Since \(\int 2 x d x=x^{2}+C\)
And \(f(0)=2\)
So, \(C=2\)
Hence, \(f(x)=x^{2}+2\)

Q.2. Evaluate \(\int 8 \sqrt{x} d x\)
Ans: Since antiderivative of \(f(x)=\sqrt{x}\) is \(F(x)=\frac{2}{3} x^{\frac{3}{2}}\)
So, \(\int 8 \sqrt{x} d x=8 \int \sqrt{x} d x=8\left(\frac{2}{3} x^{\frac{3}{2}}+C\right)=\left(\frac{16}{3} x^{\frac{3}{2}}+k\right)\)
Where \(k\) is some other constant

Q.3. Evaluate \(\int_{1}^{2} 3 d x\)
Ans: \(\int_{1}^{2} 3 d x=\left.3 x\right|_{1} ^{2}=3(2)-3(1)=3\)

Q.4. Evaluate \(\int_{1}^{e} \frac{1}{x} d x\)
Ans: \(\int_{1}^{e} \frac{1}{x} d x=\left.\ln x\right|_{1} ^{e}= \ln (e)- \ln (1)=1-0=1\)

Q.5. Evaluate \(\int_{0}^{\frac{\pi}{2}} \sin x d x\)
Ans: \(\int_{0}^{\frac{\pi}{2}} \sin x d x=-\text{cosx}\vert_0^\frac\pi2=-\left(\cos \left(\frac{\pi}{2}\right)-\cos (0)\right)=1\)

Summary

Integral calculus is the branch of calculus where we deal with the theory, properties and applications of Integral. Definite Integral and indefinite integral are two useful categories that are used to solve many real-life problems and the fundamental theorem of integral calculus connects the derivative and the integral.

We learnt about how Integral calculus is used in every area of our lives to assist us in solving problems, forecasting the future, exploring the unknown, and comprehending our world and universe.

Frequently Asked Questions

Q.1. Who is the father of integral calculus?
Ans: Isaac Newton (and also Gottfried Wilhelm Leibniz) is considered to be the father of integral calculus.

Q.2. Why is calculus so hard?
Ans: Calculus isn’t always intuitive. It requires thinking about mathematics in terms of movement in multiple directions. It also requires thinking in terms of infinite and infinitesimals, most people think more concretely than abstractly, which makes Calculus hard.

Q.3. How do you solve an integral in calculus?
Ans:  We solve an integral by applying the methods of integration. Basic and derived formulae are used to solve an integral.

Q.4. What is integral calculus and what is its process?
Ans: Integral calculus is the branch of calculus where we learn about the theory, properties, and applications of Integral. It is closely related to differential Calculus and together leads to the foundation of mathematical analysis.

Q.5. Is integral calculus hard?
Ans: Integral calculus is not so difficult. To have command of this, you need to devote sufficient time and practice to comprehend basic concepts. It involves critical thinking sometimes. You can have command on this if you have cleared your basic concepts.

Q.6. What are the different parts of calculus?
Ans: Differential and integral calculus are two parts of calculus.

Q.7. Who is the father of integral calculus?
Ans: Isaac Newton (and also Gottfried Wilhelm Leibniz) is considered to be the father of integral calculus.

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