![]() Solve the line integral over the desired curvy path to calculate the required area. Ĭalculate ‘ds’ after obtaining the parametric form. So, it first needs to be translated into its parametric form x(t), y(t), and z(t).įor example, the equation of a circle is given as, The given equation would be a function of x, y, and z. ![]() If it is a problem involving the work done on an object, then f(x,y,z) represents the force on the object.ĭetermine its parametric equations which are represented as x(t), y(t), z(t). Identify the function f(x,y,z) in the given function and the curve ‘C’ over which the integration will take place. Step-by-Step Guide to solving Line Integrals It is also used to calculate the magnetic field around a conductor when using Ampere’s law.Īmpere’s law states that the line integral of a magnetic field B around a closed path ‘C’ is equal to the total current flowing through the area bounded by boundary ‘C’. ![]() In chemistry, line integrals are used to determine the rate of reaction and know some necessary information regarding radioactive decay reactions.Ī line integral can also be used to calculate the mass of a wire, its moment of inertia, and the center of mass of the wire. They are also used in statistics to evaluate survey data and help draw out useful strategies. Line integrals are also used to find the velocity and trajectory of an object, predict the position of the planets, and understand electromagnetism in depth. While launching exploratory satellites, they consider the path of the different orbiting velocities of Earth and the planet the probe is targeted for. Space flight engineers regularly use line integrals for long missions. In electrical engineering, line integrals are used to determine the exact length of power cable needed to connect two substations that could be miles away from each other. So, a line integral over his route will help to determine the total work done or calories that a swimmer will burn in swimming along the desired path. The total work that he needs to do would vary upon the strength and direction of the current. In the field of classical mechanics, line integrals are used to calculate the work done by an object of mass m, moving in a gravitational field.Īlso, if one wants to figure out how many calories a swimmer might burn in swimming along a certain route, provided the currents in all areas can be accurately predicted. It is an extension of simple integrals and is mostly applicable for curvy surfaces. Line integrals can be used to find the three-dimensional surface areas. The line integral of the vector field is also interpreted as the amount of work that a force field does on a particle as it moves along a curve. These vector-valued functions are the ones whose input and output size are similar and we usually define them as vector fields. We can also incorporate certain types of vector-valued functions along a curve. One can also incorporate a scalar-value function along a curve, obtaining such as the mass of wire from its density. The value of the vector line integral can be evaluated by summing up all the values of the points on the vector field.Ī line integral (also known as path integral) is an integral of some function along with a curve. We can integrate both scalar-valued function and vector-valued function along a curve. The function which is to be integrated can either be represented as a scalar field or vector field. ![]() (Image will be uploaded soon) Line Integral DefinitionĪ line integral is an integral in which a function is integrated along some curve in the coordinate system. In this article, we will study a line integral, line integral of a vector field, line integral formulas, etc. In classical mechanics, line integral is used to compute the word performed on mass m moving in a gravitational field. Line integrals have several applications such as in electromagnetic, line integral is used to estimate the work done on a charged particle travelling along some curve in a force field defined by a vector field. A line integral is also known as a path integral, curvilinear integral, or curve integral. In calculus, a line integral is represented as an integral in which a function is to be integrated along a curve. ![]()
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