definite integral

Examples of definite integral in the following topics:

• Approximate Integration

  • The trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral $\int_{a}^{b} f(x)\, dx$.
  • Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral.
  • These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets.
  • The trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral $\int_{a}^{b} f(x)\,dx$.
  • Use the trapezoidal rule to approximate the value of a definite integral

• Numerical Integration

  • Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.
  • Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and, by extension, the term is also sometimes used to describe the numerical solution of differential equations.
  • This article focuses on calculation of definite integrals.
  • The basic problem considered by numerical integration is to compute an approximate solution to a definite integral:
  • If $f(x)$ is a smooth well-behaved function, integrated over a small number of dimensions and the limits of integration are bounded, there are many methods of approximating the integral with arbitrary precision.

• Improper Integrals

  • An Improper integral is the limit of a definite integral as an endpoint of the integral interval approaches either a real number or $\infty$ or $-\infty$.
  • An improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or $\infty$ or $-\infty$ or, in some cases, as both endpoints approach limits.
  • Such an integral is often written symbolically just like a standard definite integral, perhaps with infinity as a limit of integration.
  • The narrow definition of the Riemann integral also does not cover the function $\frac{1}{\sqrt{x}}$ on the interval $[0, 1]$.
  • The problem here is that the integrand is unbounded in the domain of integration (the definition requires that both the domain of integration and the integrand be bounded).

• The Definite Integral

  • A definite integral is the area of the region in the $xy$-plane bound by the graph of $f$, the $x$-axis, and the vertical lines $x=a$ and $x=b$.
  • Integrals such as these are termed definite integrals.
  • Definite integrals appear in many practical situations.
  • A definite integral of a function can be represented as the signed area of the region bounded by its graph.
  • Compute the definite integral of a function over a set interval

• Numerical Integration

  • Numerical integration is a method of approximating the value of a definite integral.
  • These integrals are termed "definite integrals."
  • Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.
  • A definite integral of a function can be represented as the signed area of the region bounded by its graph.
  • Solve for the definite integral of a continuous function over a closed interval

• Basic Integration Principles

  • Given a function $f$ of a real variable $x$, and an interval $[a, b]$ of the real line, the definite integral $\int_a^b \!
  • More rigorously, once an anti-derivative $F$ of $f$ is known for a continuous real-valued function $f$ defined on a closed interval $[a, b]$, the definite integral of $f$ over that interval is given by
  • The integral of a linear combination is the linear combination of the integrals.
  • If we are going to use integration by substitution to calculate a definite integral, we must change the upper and lower bounds of integration accordingly.
  • A definite integral of a function can be represented as the signed area of the region bounded by its graph.

• Antiderivatives

  • Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
  • Antiderivatives are important because they can be used to compute definite integrals with the fundamental theorem of calculus: if $F$ is an antiderivative of the integrable function $f$, and $f$ is continuous over the interval $[a, b]$, then
  • Because of this rule, each of the infinitely many antiderivatives of a given function $f$ is sometimes called the "general integral" or "indefinite integral" of $f$, and is written using the integral symbol with no bounds:
  • $C$ is called the arbitrary constant of integration.
  • Calculate the antiderivative (aka the indefinite integral) for a given function

• Double Integrals Over Rectangles

  • The multiple integral is a type of definite integral extended to functions of more than one real variable—for example, $f(x, y)$ or $f(x, y, z)$.
  • Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the $x$-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three dimensional Cartesian plane where $z = f(x, y))$ and the plane which contains its domain.
  • Formulating the double integral , we first evaluate the inner integral with respect to $x$:
  • We could have computed the double integral starting from the integration over $y$.
  • Double integral as volume under a surface $z = x^2 − y^2$.

• Integration Using Tables and Computers

  • Tables of known integrals or computer programs are commonly used for integration.
  • Integration is the basic operation in integral calculus.
  • A compilation of a list of integrals and techniques of integral calculus was published by the German mathematician Meyer Hirsch as early as in 1810.
  • Computers may be used for integration in two primary ways.
  • First, numerical methods using computers can be helpful in evaluating a definite integral.

• Surface Integrals of Vector Fields

  • The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field.
  • The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
  • Alternatively, if we integrate the normal component of the vector field, the result is a scalar.
  • This formula defines the integral on the left (note the dot and the vector notation for the surface element).
  • Explain relationship between surface integral of vector fields and surface integral of a scalar field