integration

Examples of integration in the following topics:

• 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 $-\infty$.
  • Such an integral is often written symbolically just like a standard definite integral, perhaps with infinity as a limit of integration.
  • It is often necessary to use improper integrals in order to compute a value for integrals which may not exist in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function, or an infinite endpoint of the domain of integration.
  • However, the Riemann integral can often be extended by continuity, by defining the improper integral instead as a limit:
  • Evaluate improper integrals with infinite limits of integration and infinite discontinuity

• Iterated Integrals

  • An iterated integral is the result of applying integrals to a function of more than one variable.
  • An iterated integral is the result of applying integrals to a function of more than one variable (for example $f(x,y)$ or $f(x,y,z)$) in such a way that each of the integrals considers some of the variables as given constants.
  • If this is done, the result is the iterated integral:
  • Similarly for the second integral, we would introduce a "constant" function of $x$, because we have integrated with respect to $y$.
  • Use iterated integrals to integrate a function with more than one variable

• Line Integrals

  • A line integral is an integral where the function to be integrated is evaluated along a curve.
  • A line integral (sometimes called a path integral, contour integral, or curve integral) is an integral where the function to be integrated is evaluated along a curve.
  • The function to be integrated may be a scalar field or a vector field.
  • This weighting distinguishes the line integral from simpler integrals defined on intervals.
  • The line integral finds the work done on an object moving through an electric or gravitational field, for example.

• 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 $f(x, y, z)$.
  • 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 $z = x^2 − y^2$.
  • The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated.

• Change of Variables

  • One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae.
  • The limits of integration are often not easily interchangeable (without normality or with complex formulae to integrate).
  • One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae.
  • When changing integration variables, however, make sure that the integral domain also changes accordingly.
  • Use a change a variables to rewrite an integral in a more familiar region

• 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.
  • We also may have to resort to computers to perform an integral.
  • 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.

• Numerical Integration

  • Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.
  • 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.
  • There are several reasons for carrying out numerical integration.

• Integrative Psychotherapy

  • In contrast, integrative psychotherapy attends to the relationship between theory and technique.
  • In contrast, an integrative therapist is curious about the "why and how" as well.
  • There are many approaches to integrating psychotherapeutic techniques.
  • Theoretical integration: This approach requires integrating theoretical concepts from different approaches.
  • Assimilative integration: This mode of integration favors a firm grounding in any one system of psychotherapy, but with a willingness to incorporate or assimilate, perspectives or practices from other schools.

• 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.
  • Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral.
  • The integrand is evaluated at a finite set of points called integration points and a weighted sum of these values is used to approximate the integral.

• Volumes

  • Volumes of complicated shapes can be calculated using integral calculus if a formula exists for the shape's boundary.
  • A volume integral is a triple integral of the constant function $1$, which gives the volume of the region $D$.
  • Using the triple integral given above, the volume is equal to:
  • Triple integral of a constant function $1$ over the shaded region gives the volume.
  • Calculate the volume of a shape by using the triple integral of the constant function 1