Differentiation and Integration both are fundamental concepts of Calculus, which no doubt is among the outstanding achievements of human knowledge. The concepts and implications of Calculus have helped man in many disciplines from astronomy, economics, history, and physics to even biology.
And the basic
principles of calculus implemented in all these broad fields are
differentiation and integration. In relation to the application of these
fundamental concepts every student no matter from which discipline is, must be
aware of the basis of integration and differentiation.
In this article,
we would discuss why integration and differentiation are important for students
to learn.
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Integration
The term integration
has a simple definition, “the process of finding out the integrals”. While an
integral is a term that allocates the numbers to functions while describing
units that form as a result of joining minute quantities. The integrals are
classified into two subcategories
· Indefinite Integrals
· Definite Integrals
The Definite integrals
are simply used to find the area under the curve while the indefinite integrals
are a function the derivative of which is a given function or also called the
antiderivatives. Together both these integrals serve to solve several problems
of calculus. You can find the Indefinite Integrals and Definite Integrals
values with the help of online tools like definite integral calculator and indefinite integral calculator.
However, the main
difference between both the terms is, the definite integral in its existence,
is a real number value, while on the other hand antiderivative has an unlimited
number of functions that vary just by a constant.
The best thing about
the integration process is that most of its formulas could be directly derived
from their corresponding derivative formulas. While a few are there including
integration by parts, trigonometric substitution, change and substitution of
variables, and trigonometric variables whose formula needs to be derived by
expending some extra calculations and derivations.
The integration helps
us in finding the following real-life quantities:
Area between curves
Distance, Velocity, Acceleration
Volume
Average value of a function
Work
Center of Mass
Kinetic energy; improper integrals
Probability
Arc Length
Surface Area
- Differentiation
Finding out the
function of the rate of change of any variable in relation to another is known
as the process of differentiation. Generally, the integration process is used
to find the rate of change for instance if we need to find the rate of change
of velocity (acceleration) with respect to time we require to implement the
rules of differentiation.
If we explain it in
terms of mathematics that implies the process of finding the rate of change of
x with respect to y is the differentiation. This process requires several rules
to differentiate these functions. Thus we can say that finding the derivative
for any function is known as a different ion.
While the derivative calculates the change in the output (function) value in relation to the change in the input value. For instance, to calculate how rapidly an object changes its position with an increase in time we would say in terms of derivatives calculating its velocity. Derivatives are implemented in vast disciplines of physical sciences and the basic concept of calculus.
When a derivative of a single-variable function occurs at a given input value, it is the slope of the tangent line to the function's graph at that point. The tangent line is the function's best linear approximation near the input value.
As a result, the
derivative is often referred to as the "instantaneous rate of
change," or the ratio of the dependent variable's instantaneous change to
that of the independent variable. Also, a constant function would always have a
zero as a derivative.
In real life
differentiation help us in finding the rate of change of the following
quantities:
The rate of change in velocity with time, the acceleration
Finding the tangent and normal in a curve
Finding out the turning point
Calculating the highest and lowest point in a curve or graph
The main
differentiation rules are:
· Sum and Difference
Rule
· Product Rule·
. Quotient Rule
· Chain Rule
The article is produced by Sarah Taylor, one of the guest contributors to the School of Literature.
