vertical integration

(noun)

The integrating of successive stages in the production and marketing process under the ownership or control of a single management organization.

Related Terms

Examples of vertical integration in the following topics:

  • Growth through buying out other companies

    • Firms can also integrate vertically through acquisition, i.e. by buying out suppliers or customers to process more steps in the value chain in-house.
    • Vertical integration can sometimes bring advantages of cost or differentiation.
    • Cost advantages can arise either through buying or building up cheaper distribution channels (forward integration), or cheap inputs (backward integration).

  • 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.
    • An improper Riemann integral of the second kind.The integral may fail to exist because of a vertical asymptote in the function.
    • Evaluate improper integrals with infinite limits of integration and infinite discontinuity

  • Cylindrical Shells

    • Shell integration (also called the shell method) is a means of calculating the volume of a solid of revolution when integrating perpendicular to the axis of revolution .
    • (When integrating parallel to the axis of revolution, you should use the disk method. ) While less intuitive than disk integration, it usually produces simpler integrals.
    • Integration, as an accumulative process, can then calculate the integrated volume of a "family" of shells (a shell being the outer edge of a hollow cylinder), giving us the total volume.
    • Each segment located at $x$, between $f(x)$and the $x$-axis, gives a cylindrical shell after revolution around the vertical axis.
    • Use shell integration to create a cylindrical shell and calculate the volume of a "solid of revolution" perpendicular to the axis of revolution.

  • Area of a Surface of Revolution

    • If the curve is described by the function $y = f(x) (a≤x≤b)$, the area $A_y$ is given by the integral $A_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$ for revolution around the $x$-axis.
    • If the curve is described by the parametric functions $x(t)$, $y(t)$, with $t$ ranging over some interval $[a,b]$ and the axis of revolution the $y$-axis, then the area $A_y$ is given by the integral:
    • If the curve is described by the function $y = f(x)$, $a \leq x \leq b$, then the integral becomes:
    • A portion of the curve $x=2+\cos z$ rotated around the $z$-axis (vertical in the figure).
    • Use integration to find the area of a surface of revolution

  • Indefinite Integrals and the Net Change Theorem

    • An indefinite integral is defined as $\int f(x)dx = F(x)+ C$, where $F$ satisfies $F'(x) = f(x)$ and where $C$ is any constant.
    • $f(x)$, the function being integrated, is known as the integrand.
    • Note that the indefinite integral yields a family of functions.
    • Essentially, the graphs of antiderivatives of a given function are vertical translations of each other, with each graph's location depending upon the value of $C$.
    • Apply the basic properties of indefinite integrals, including the constant, sum, and difference rules

  • Antiderivatives

    • Essentially, the graphs of antiderivatives of a given function are vertical translations of each other, with each graph's location depending upon the value of $C$.
    • 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

  • Calculus with Parametric Curves

    • The position of the object is given by $x$ and $y$, signifying horizontal and vertical displacement, respectively.
    • This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves term-wise.
    • This makes integration and differentiation easier to carry out as they rely on the same variable.
    • Writing $x$ and $y$ explicitly in terms of $t$ enables one to differentiate and integrate with respect to $t$.
    • Use differentiation to describe the vertical and horizontal rates of change in terms of $t$

  • 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 $x=b$.
    • Given a function $f$ of a real variable x and an interval $[a, b]$ of the real line, the definite integral $\int_{a}^{b}f(x)dx$ is defined informally to be 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 $x=b$, such that the area above the $x$-axis adds to the total, and that the area below the $x$-axis subtracts from the total.
    • Integrals such as these are termed definite integrals.
    • Definite integrals appear in many practical situations.
    • The notation for this integral will be:

  • Numerical Integration

    • Numerical integration is a method of approximating the value of a definite integral.
    • Given a function $f$ of a real variable $x$ and an interval of the real line, the definite integral $\int_{a}^{b}f(x)dx $ is defined informally to be 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 $x=b$, such that the area above the $x$-axis adds to the total, and that the area below the $x$-axis subtracts from the total.
    • 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.

  • Basic Integration Principles

    • f(x)\,dx$ is defined informally to be the area of the region in the $xy$-plane bounded by the graph of $f$, the $x$-axis, and the vertical lines $x=a$ and $x=b$, such that area above the $x$-axis adds to the total, and that below the $x$-axis subtracts from the total.
    • The collection of all anti-derivatives is called the indefinite integral of $f$ and is written as
    • The integral of a linear combination is the linear combination of the integrals.
    • By reversing the chain rule, we obtain the technique called integration by substitution.
    • 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.