line integral

(noun)

An integral the domain of whose integrand is a curve.

Related Terms

Examples of line integral in the following topics:

  • 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.
    • 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.
    • For some scalar field $f:U \subseteq R^n \to R$, the line integral along a piecewise smooth curve $C \subset U$ is defined as:

  • Green's Theorem

    • Green's theorem gives relationship between a line integral around closed curve $C$ and a double integral over plane region $D$ bounded by $C$.
    • Green's theorem gives the relationship between a line integral around a simple closed curve $C$ and a double integral over the plane region $D$ bounded by $C$.
    • In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area.
    • In plane geometry and area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.
    • Green's theorem can be used to compute area by line integral.

  • Fundamental Theorem for Line Integrals

    • Gradient theorem says that a line integral through a gradient field can be evaluated from the field values at the endpoints of the curve.
    • The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
    • The gradient theorem implies that line integrals through irrotational vector fields are path-independent.
    • where the definition of the line integral is used in the first equality and the fundamental theorem of calculus is used in the third equality.
    • Electric field lines emanating from a point where positive electric charge is suspended over a negatively charged infinite sheet.

  • Stokes' Theorem

    • Stokes' theorem relates the integral of the curl of a vector field over a surface to the line integral of the field around the boundary.
    • The generalized Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
    • Stokes' theorem says that the integral of a differential form $\omega$ over the boundary of some orientable manifold $\Omega$ is equal to the integral of its exterior derivative $d\omega$ over the whole of $\Omega$, i.e.:
    • Given a vector field, the theorem relates the integral of the curl of the vector field over some surface to the line integral of the vector field around the boundary of the surface.

  • Area and Arc Length in Polar Coordinates

    • Area and arc length are calculated in polar coordinates by means of integration.
    • If you were to straighten a curved line out, the measured length would be the arc length.
    • The length of $L$ is given by the following integral:
    • An infinite sum of these sectors is the same as integration.
    • The curved lines bounding the region $R$ are arcs.

  • Basic Integration Principles

    • Given a function $f$ of a real variable $x$, and an interval $[a, b]$ of the real line, the definite integral $\int_a^b \!
    • f(x)\,dx$ 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 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.

  • Integration By Parts

    • Integration by parts is a way of integrating complex functions by breaking them down into separate parts and integrating them individually.
    • In calculus, integration by parts is a theorem that relates the integral of a product of functions to the integral of their derivative and anti-derivative.
    • The rule can be derived in one line by simply integrating the product rule of differentiation.
    • Now, let's take a look at the principle of integration by parts:
    • Assuming the curve is smooth within a neighborhood, this generalizes to indefinite integrals $\int xdy + \int y dx = xy$, which can be rearranged to the form of the theorem: $\int xdy = xy - \int y dx$.

  • 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.

  • Volumes

    • One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in three-dimensional space.
    • 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

  • 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.
    • The volume of the solid formed by rotating the area between the curves of $f(x)$ and $g(x)$ and the lines $x=a$ and $x=b$ about the $y$-axis is given by:
    • The volume of solid formed by rotating the area between the curves of $f(y)$ and and the lines $y=a$ and $y=b$ about the $x$-axis is given by:
    • Use shell integration to create a cylindrical shell and calculate the volume of a "solid of revolution" perpendicular to the axis of revolution.