Integral calculus is a mathematical field that studies the idea of integration. Integration is the process of finding the antiderivative or the integral of a function. It is the inverse operation of differentiation, which includes determining a function’s derivative.
Integration has several practical applications in physics, engineering, economics, and other domains. It allows us to find areas, volumes, displacement, work done, and various other quantities that can be represented as the accumulation of smaller parts.
In this article, we have discussed the formulas, definition of integral in calculus, rules of types of integral in calculus, and application of integral. Also, we discussed the topic with the help of detailed example of the integral of calculus.
Definition
Integral is a mathematical operation that finds the area under a curve or the accumulation of a quantity over a given interval. The integral of a function f(x) is represented as f(x)dx, where f(x) is the integrand and dx is the variable of integration (typically x, although it might be any variable).
The result of the integration is typically expressed as an indefinite integral, which includes an arbitrary constant of integration (C).
Formula of Integral
The integral of a function f(x) concerning the variable x is denoted as ∫f(x)dx. The formula for finding integral are:
Integral: Types
Definite and indefinite there are two main types of integral in calculus now we discuss them one by one here
Definite integral
In definite integral, both limits are given. The definite integral calculates the net area between the curve of a function and the x-axis over a specified interval [a, b]. It gives a numerical value, not a function.
Example:
∫31 (2x + 3) dx = | (x2 + 3x) |31
(32 + 3×3) – (12 + 3×1) = 14
Indefinite Integral:
The indefinite integral represents a family of functions whose derivatives are the original function f(x).
It is written as
Example:
∫(2x + 3) dx = x2 + 3x + C
Integral in calculus: Rules
In calculus, there are several rules and techniques for evaluating integrals.
Here, we discuss some basic rules of integration to solve the function
| Rule name | Expression | Formula |
| Power Rule | ∫xn dx | x(n+1)/(n+1) + C |
| Constant Rule | ∫c dx | cx + C |
| Sum and Difference Rule | ∫[f(x) + g(x)] dx∫[f(x) – g(x)] dx | ∫f(x) dx + ∫g(x) dx∫f(x) dx – ∫g(x) dx |
| Constant Multiple Rule | ∫c× f(x) dx | c× ∫f(x) dx |
| Exponential Rule | ∫ex dx | ex + C |
| Trigonometric Rules | ∫sin(x) dx = ∫cos(x) dx = ∫sec2(x) dx = ∫csc2(x) dx = ∫sec(x)tan(x)dx =∫csc(x)cot(x) dx = | -cos(x) + Csin(x) + Ctan(x) + C-cot(x) + Csec(x) + C-csc(x) + C |
| Integration by Parts | ∫u dv = | uv – ∫v du |
| Substitution Rule | ∫f(g(x)) g'(x) dx = | ∫f(u) du |
These rules, along with various techniques and methods, help in solving a wide range of integrals. However, integration can be quite challenging and may require creativity and practice to apply the right rules in different situations.
Application of integral
Integral in Calculus has many applications. Here, we discuss some main applications of integral calculus are
1. Calculating Areas and Volumes: Integration allows us to find the area under a curve, the area between two curves, and the volume of a solid obtained by revolving a curve around a particular axis.
2. Physics and Engineering: Integration is used to analyze physical quantities like velocity, acceleration, displacement, and force. For example, integrating acceleration gives velocity, and integrating velocity gives the displacement.
3. Calculating Work and Energy: Integration is employed to determine work done by a force, the amount of energy stored in a system, and to calculate potential energy and kinetic energy.
4. Probability and Statistics: In probability theory, integration is used to compute probabilities, cumulative distribution functions, and expected values of random variables.
5. Economics and Finance: Integration is used in economics to calculate consumer surplus and producer surplus, and in finance to evaluate present value, future value, and annuity calculations.
6. Signal Processing: In signal processing, integration is utilized to compute the cumulative sum of signals and to analyze the behavior of continuous signals.
7. Electrical Engineering: Integration is applied in electrical circuits to determine charge, voltage, and current in various components.
8. Fluid Mechanics: Integration is used to calculate fluid flow rates, pressure distribution, and various fluid-related parameters.
9. Computer Graphics: In computer graphics, integration helps to render images, calculate lighting effects, and simulate physical phenomena in virtual environments.
How to find the integral of the function?
Example number 1:
F(x)= sin(x)-cos(x)
∫ F(x)dx=?
Solution:
Step 1: Split the integral
∫f(x)dx =
- ∫sin(x) dx – ∫cos(x) dx
Step nb2:
Evaluate the integral of each term
∫sin(x) dx=-cos(x) + C1
And
∫cos(x) dx=sin(x) + C2,
Step 3: Combine the results
∫ (sin(x) – cos(x)) dx =
- ∫sin(x) dx – ∫cos(x) dx
- (-cos(x) + C1) – (sin(x) + C2)
- -cos(x) + C1 – sin(x) – C2
After combining all constant results give the single constant result (C = C1 – C2):
∫ (sin(x) – cos(x)) dx =
- -cos(x) – sin(x) + C
Example number 2:
∫ (3x2+8x-1) dx=?
Solution:
Given function
∫ (3x2+8x-1) dx=?
Determine the integral of a given function
Step 1: In the first step apply the integral on all function
∫ (3x2+8x-1) dx=3∫ x2 dx+8∫x dx-∫1dx (1)
Step 2:
Evaluate the integral of each term
3∫ x2 dx
By using the power rule
3∫ x2 dx = 3x3/3 = x3
8∫x dx
Again, using the power rule
8∫ x dx = 8x2/2 + C
=4x2+c
∫1dx
By using constant rule
x+C
Step 3: Combine the results
∫ (3x2 +8x-1) dx= x3+4x2+c-x- C
We know that all constant which is equal to the single constant
∫ (3x2 +8x-1) dx= x3+4x2+c-x- C
∫ (3x2 +8x-1) dx= x3+4x2-x+ C
You can also use online tools like antiderivativecalculator.net to find the step-by-step solution of integral problems.
FAQs of integral
Question 1:
How is the integral represented?
Answer:
The integral of a function f(x) concerning x is represented as f(x) dx. The symbol ∫ represents the integral, and dx indicates that we are integrating concerning the variable x.
Question 2:
It is possible to integrate all functions?
Answer:
While the integrals of most elementary functions may be defined in terms of elementary functions, not all functions have closed-form solutions. Integrals cannot always be represented using ordinary functions, hence numerical approaches or approximations are needed.
Question number 3:
What is the determination of definite integrals?
Answer:
Definite integrals are used to compute aggregate values like as the total area under a curve, the total distance traveled by an item, the total charge in an electric field, or the average value of a function during a certain time.
Summary
In this article, we have discussed the formulas, definition of integral in calculus, rules of types of integral in calculus, and application of integral. Also, we discussed the topic with the help of detailed example of the integral of calculus. I hope you will be able to defend this topic easily after completely understanding this article.
