Concepts of Integration and Derivatives in Calculus

An introduction

Integration and Derivatives both are the basic concepts of calculus in Math that deals with changes. Almost every student that is pursuing a course in Mathematics or Calculus alone must be aware of the terms. But only a few students are there who truly understand the concept.

The main thing to be considered, however, is that they both are inverse functions of each other. In other words, we can say that both the terms are analogous to each other. But how are they inverse to each other? The simple explanation for this is, derivatives find the slopes while integration finds the areas or volume.

In terms of Mathematics, we can say that the product of “a” and “b” is the area whereas the ratio of “y/x” is the slope. These ratios and products are inverses of each other so as Integration and Derivative.

Both the integration and derivatives have their scope and implication in other subjects as well including economics, physics, and history. Furthermore, the growth rate could also be found using concepts of derivatives.

In this article, we should go through the key concepts of integration and derivatives that every student must learn while studying calculus.

Integration

Integration is basically the process involved in finding the indefinite integrals in calculus. The process of integration utilizes various properties to find the area under the curve by a function. In Maths integrals play an important role in finding the quantities including area, volume displacement, that can’t be singularly calculated.

Generally, while discussing integrals one is discussing definite integrals, which are used for antiderivatives. The indefinite integral can never be unique as differentiation is because the constant’s derivative is always zero. Calculatored’s integration calculator with steps is used to solve integral questions so they are very helpful in understanding integrals.

In calculus, wherever geometry and algebra are being applied the concept of limit is utilized. And this limit facilitates the evaluation of the functions between the two points on a graph. For instance, the limit calculates if the distance between two points is nearly zero how these two points could get nearer to each other.

Major types of Integrals

The infinite integral defines a family of parallel curves with parallel tangents at the intersection points of any curve in the family, with lines orthogonal to the axis defining the integration component. Also, the integral of a function is taken all over an interval in integrals. For ease, an integral calculator with steps is used for solving questions of integration.

Integrals are classified into two major classes, namely:

• Definite Integrals
• Indefinite Integrals

Definite integral as shown by its name that it will provide the definite result or in the easier way we can say that, definite integral is used to evaluate the area under the curve within the exact range of limits. These are called the upper limit and lower limit under which we have to find the area.

Moreover, the Definite integral always results in the exact value in real numbers.

Just opposite to that indefinite integral is used to find the function of the area finding under the curve. In indefinite integral, we cannot get the exact value of the area which have been evaluated. It is visually represented as integral than a function which we have to integrate and dx or dy about which we have to integrate that function.

Applications of integration

Integration is used in various different fields in our daily life. Some of the important uses of integration in a real-life are given as follow:

• Integration is used to find the area between the curves.
• It is also used to find the distance, speed, velocity or acceleration.
• Integrals are evaluated to find the center of mass.
• It also has an application while dealing in Kinetic energy

Derivatives

A function’s changing rate is called a derivative whereas the derivative of an independent variable value is the differentiation. The derivative is used to find the tangent of a function at a particular point. But have we ever thought about the implication of derivatives in real life?

Derivatives are not just mathematical concepts they do have in real life as they are used to find the smallest change in any quantity if a unit change occurs in another quantity. As mentioned above, differentiation is the major concept or essence of calculus and it is used in calculus to determine changes that take place per unit in an independent variable.

Every function’s derivative is always unique in calculus. Also, the differentiation process has a unique property which is termed linearity. Utilizing the product and addition properties, linearity makes the derivatives more natural for functions by constant.

Derivative as a Slope of Function

Derivative, the slope of a function is always taken at a specific point while calculating. While in terms of geometry the rate of change of a quantity with respect to another quantity is described by the derivative of a function. The differentiate calculator is also proving to be very helpful for solving derivation online without having any complex brainstorming.

Major Rules of Differentiation

Some of the major rules of differentiation include:

• Power Rule

• Sum and Difference Rule

• Product Rule

• Quotient Rule

• Chain Rule

Applications of Differentiation

As we talk derivatives are just bound with integrals but in an antiparallel direction. where there are applications of integrals there are also different uses of derivatives in our life. Some of them are enlisted below:

• Derivatives are used in the analysis of profit or loss in the business.
• It is also used for calculating speed or distance covered e.g miles/hr or km/hr.
• Differentiation is used to find the temperature variation.
• Derivatives are used to derive many equations in physics.
• Derivatives have applications while approximation.
• It is also helpful while finding the maximum and minimum values.

Conclusion

Integration and differentiation are inverse processes to each other. Well, in short, derivatives are helpful to calculate the gradient of the curve. Derivatives use to check the instantaneous rate of change for a very small instant. To find the area under or between the curves we will always use integration. No doubt these two concepts discover after a long pause but these have interrelated to each other.