Line Integral

A line integral enables the computation of the area of a three-dimensional surface. Line integrals are useful in a wide range of situations. They may be used to determine the work done by a charged particle moving along a curve in a force field represented by a vector field, for example. A line integral is a calculus term that refers to an integral in which the function being integrated is evaluated along a curve. The term “line integral” is also used to refer to the route integral, curve integral, or curvilinear integral. This article will examine the definition of a line integral, its formulae, examples, and applications in real life.

Definition of Line Integral

By this point, you should be familiar with integral construction. We decompose a geometrical form into small parts, multiply the component’s size by the function value associated with that piece, and then sum all the results. The geometrical figure for one variable integration is a line segment, for double integration, it is a region, and for triple integration, it is solid.

Today’s geometrical figure is a curve. If a curve has a function specified on it, we can decompose the curve into small line segments, multiply the length of the line segments by the function value on the segment, and sum the products. When is customary, we shall impose a restriction as the line segments’ length approaches zero. The term “line integral” refers to this new number, which may be specified in two, three, or higher dimensions.

Assume that the density of a wire is f(x,y,z)f(x,y,z) at the position (x,y,z)(x,y,z). Then the line integral equals the wire’s entire mass. The following is a symbol-based definition.

Let f be a function defined on a finite-length curve C. If this is the case, then the line integral of f along C is

∫Cf(x,y)ds=limn→∞∑i=1nf(xi,yi)Δsi

(for two-dimensional objects)

∫Cf(x,y,z)ds=limn→∞∑i=1nf(xi,yi,zi)Δsi

(for three-dimensional objects)

Each point in space has a value associated with it in a scalar field. Scalar fields include height, temperature, and pressure maps. The value at each point in a two-dimensional field may be thought of as the height of a surface embedded in three dimensions. The line integral of a curve passing through this scalar field is equal to the area beneath a curve drawn over the field’s surface.

Line Integral Formula

The Line Integral of a Scalar Field is represented by the following formula:

For a scalar field with function f: U ⊆ Rn → R, a line integral along with a smooth curve, C ⊂ U is defined as:

∫C f(r) ds =

∫ab

f[r(t)] |r’(t)| dt

Here, r: [a, b]→C is an arbitrary bijective parametrization of the curve.

r (a) and r(b) gives the endpoints of C and a < b.

The Line Integral of a Vector Field is represented by the following formula:

For a vector field with function, F: U ⊆ Rn → Rn, a line integral along with a smooth curve C ⊂ U, in the direction “r” is defined as:

∫C F(r). dr =

∫ab

F[r(t)] . r’(t)dt.

Here, “.” represents the dot product.

Line Integral Applications

The line integral is useful in a variety of situations. The surface area of three-dimensional planes is calculated using a line integral. Several applications of line integrals in vector calculus include the following:

• The mass of the wire is calculated using a line integral.
• It aids in the calculation of the wire’s moment of inertia and centre of mass.
• It is used to calculate the magnetic field surrounding a conductor in Ampere’s Law.
• A line integral is used in Faraday’s Law of Magnetic Induction to determine the voltage generated in a loop.
• The line integral may be used to determine the amount of work performed by a force on a moving item in a vector field.

Conclusion

A line integral enables the computation of the area of a three-dimensional surface. Line integrals are useful in a wide range of situations. They may be used to determine the work done by a charged particle moving along a curve in a force field represented by a vector field, for example.