Integrals Definitions and Examples
Introduction
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the fundamental operations of calculus and its inverse, differentiation, is one of the most important techniques in analysis. The word “integral” comes from the Latin word integr?re, which means “to make whole”.
What is Integral Calculus?
Integral calculus is the study of integrals, which are mathematical objects that can be thought of as generalizations of summation. In integral calculus, we study how to calculate these objects and what they can tell us about the underlying functions.
There are two types of integrals: definite and indefinite. A definite integral has a fixed boundaries, while an indefinite integral does not. indefinite integrals are often used to find the area under a curve, while definite integrals can be used to calculate things like volume or displacement.
Integrals can be calculated using a variety of methods, including integration by substitution, integration by parts, and numerical integration. The choice of method depends on the particular problem being solved.
In addition to providing a way to calculate important quantities, integrals also give us insight into the behavior of functions. For example, the mean value theorem tells us that if we take an integral over a function f(x), then the average value of f(x) over that interval will be equal to the value of the function at the midpoint of the interval. This theorem can be used to prove a variety of results about how functions behave.
Fundamental Theorems of Integral Calculus
There are three fundamental theorems of integral calculus that are essential for understanding how integrals work.
The first theorem is the Fundamental Theorem of Calculus, which states that if a function is continuous on a closed interval, then it is integrable on that interval. This theorem is what allows us to define the integral of a function as the limit of a sum.
The second theorem is the First Fundamental Theorem of Calculus, which states that if a function is continuous on an open interval, then it is differentiable on that interval. This theorem tells us that differentiation and integration are inverse operations.
The third and final theorem is the Second Fundamental Theorem of Calculus, which states that if a function is continuous on an open interval, then it is integrable on that interval. This theorem allows us to evaluate definite integrals using indefinite integrals.
First Fundamental Theorem of Integrals
The First Fundamental Theorem of Integrals states that if a function is continuous on a closed interval, then it is integrable on that interval. This theorem is the basis for many of the properties of integrals that we take for granted, such as the fact that the integral of a constant function is just the product of the constant and the length of the interval.
Second Fundamental Theorem of Integrals
The Second Fundamental Theorem of Integrals states that if a function is continuous on an interval, then it can be integrated on that interval. This theorem is also known as the First Fundamental Theorem of Calculus.
This theorem is important because it allows us to find the area under a curve. To find the area under a curve, we need to be able to integrate the function on the interval. If the function is not continuous, then we cannot integrate it and therefore we cannot find the area under the curve.
There are many applications of the Second Fundamental Theorem of Integrals in physics and engineering. For example, in physics, this theorem is used to find the work done by a force over an interval. In engineering, this theorem is used to find the displacement of a body over an interval.
Types of Integrals
Riemann Integrals:
The Riemann integral is defined as the limit of a Riemann sum. Let’s say we have a function f(x) and we want to find the area under its curve between x=a and x=b. We can divide the interval [a,b] into n equal subintervals, each of width (b-a)/n. We can then approximate the area under the curve by adding up the areas of n rectangles, each with height f(x_i), where x_i is the ith value in our partition of [a,b]. That is, we define:
Riemann Sum = sum_{i=1}^n f(x_i) * (\frac{b-a}{n})
If we take the limit as n approaches infinity, we get the Riemann integral:
\int_a^b f(x)dx = \lim_{n \to \infty} Riemann Sum
Indefinite Integrals
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral is
The process of finding a function’s indefinite integral is called integration. Indefinite integrals are defined on a much broader class of functions than those that have definite integrals. For example, every polynomial function has a definite integral—given by the fundamental theorem of calculus—but not every polynomial function has an indefinite integral.
The most common symbols used to denote indefinite integrals are ? (the “L” symbol) and ? (the “elongated S” symbol). These symbols represent tall regions in mathematical space called “slices”.
Definite Integrals
Definite integrals are a type of integral that allow us to calculate the area under a curve. In order to calculate a definite integral, we first need to find the indefinite integral of the function we are working with. The indefinite integral will give us the general equation for the area under the curve, while the definite integral will allow us to calculate the specific value for a given set of parameters.
For example, let’s say we want to calculate the area under the curve y = x^2 from x = 0 to x = 2. We can do this by finding the indefinite integral first:
The indefinite integral of y = x^2 is ƒ(x) = ?y dx = 1/3x^3 + C
Now that we have the general equation, we can plug in our specific values to calculate the definite integral:
ƒ(2) – ƒ(0) = 1/3(2)^3 – 1/3(0)^3= 8/3 – 0= 8/3
Therefore, the area under our curve from x = 0 to x = 2 is 8/3.
Properties of Integral Calculus
Integral calculus is the branch of mathematics that deals with the finding and properties of integrals. Integrals are mathematical objects that can be thought of as a generalization of sums. They are defined in terms of a limit, and this definition gives rise to a wide variety of properties.
The most basic property of integrals is that they can be used to find the area under a curve. This property is often called the fundamental theorem of calculus. It states that if f is a continuous function on [a,b], then the definite integral
\int_a^bf(x)\,dx
is equal to the area under the curve between a and b. This theorem is the basis for many other properties and applications of integration.
Another important property of integrals is called integration by substitution. This allows one to compute an integral by making a change of variables that simplifies the problem. For example, consider the integral
\int x^3\,dx.
This can be rewritten using a substitution u = x^2 as
\int u\,du = \frac{1}{2}\int x^2\,dx = \frac{1}{2}x^4 + C.
This technique can be used to integrate a wide variety of functions.
There are also many important properties involving inverse functions and integrals. These allow one to compute integrals in terms of derivatives, which can be
Integrals Formulas
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral is
f=?abf(x)dx
The fundamental theorem of calculus states that differentiating a function produces an infinite series of integrals and reversing the process produces the original function.
Integration has applications to nearly every branch of modern mathematics, except for discrete mathematics. Integration is used to solve differential equations and to calculate areas under curves (known as integration), volumes (known as integration), surfaces (known as integration), moments (known as integration), and many other quantities.
There are three general types of integrals: indefinite integrals (which do not have defined limits), definite integrals (which do have defined limits), and improper integrals (which may or may not have defined limits). The first two are related by the fundamental theorem of calculus. The third type is related by analytic continuation.
Methods to Find Integrals
There are a few methods to find integrals, which is the process of finding the area under a curve. The most common method is integration by substitution, also known as u-substitution. This method involves solving for u in terms of x, then substituting this into the integral. Another common method is integration by parts, which involves breaking up the integral into smaller pieces that can be more easily solved. Lastly, there is the method of partial fractions, which is used when an integrand has a rational function.
Applications of Integral Calculus
Integral calculus is the study of integrals, which are mathematical functions that can be used to calculate areas, volumes, and other quantities. Integral calculus has many applications in science and engineering, such as in the calculation of motion and forces, the determination of electric potentials, and the modeling of heat transfer.
Conclusion
In conclusion, integrals are a fundamental part of mathematics that allow us to find the area under a curve and calculate volumes. Although they can be tricky to master at first, with a little practice you’ll be able to integrate like a pro!
