Line Integral
What is Line Integral in Calculus with Example?
Line Integral is the integral of a function evaluated along a line or a curve. It is also known as curve integral, path integral or curvilinear integral. For example, if we need to find work done by a force on a moving object, then it can be found be evaluating the line integral of the force field along direction of motion of the object.
What Type of Fields, Line Integral can be Calculated?
Line Integral is calculated for both types of functions or fields, i.e. scalar field and vector field. There are separate expressions to determine the value of integral in both cases.
What is Line Integral for a Scalar Field?
The expression for determining value of line integral of a scalar function is defined as follows,
∮ f(r)ds = a ∫bf[r(t)]|r'(t)|dt
What is Line Integral for a Vector Field?
The expression to determine the value of line integral of a scalar function is defined as follows,
∮ F(r) . ds = a ∫b F[r(t)] . [r'(t)]dt
What is the Difference between Line Integral and Surface Integral?
- Line integral is used to determine a function’s integral along a line or a curve.
- Surface integral is used to determine a function’s integral over a surface.
Can a Line Integral be Zero?
Yes, the value of a line integral can be zero. Its value depends upon the function being integrated and the path along which the function is integrated. Therefore, if the function is such that its components in some particular x,y and z directions cancel out each other, then line integral along that path comes out to be zero.
When Can a Line Integral be Negative?
The sign of a line integral depends on the orientation of the function and the curve along which function is to be integrated. So, when the curve is traversed in opposite direction with respect to the function to be integrated, the value of line integral comes out to be negative.
What is ‘dr’ in Line Integral?
The ‘dr’ signifies a differential displacement vector in the direction of the curve along which value of integral is to be evaluated. As a vector, dr can be expressed as <dx,dy,dz>, which represent infinitesimal displacement along coordinate axes in direction of the curve of line integral.
Line Integral
Line Integral is the integral that is evaluated along a line or a curve. Generally, an integral is calculated when we need to determine a quantity’s value such as area, volume, temperature, etc. for a larger part of the body when we have an expression for a smaller part. It is the opposite of differentiation and is also called an anti-derivative of the function. The Line Integral is used in engineering in various fields when we need to determine a function’s value along a line or along a curve. For example, if we need to calculate work done on an electron by a force field along a curve, we can determine it using line integral.
In this article, we will learn about the definition of line integral, its formula of line Integral, applications of line Integral, some solved examples based on the calculation of line integral, and some frequently asked questions related to line integral.
Table of Content
- Definition of Line Integral
- Formula of Line Integral
- For scalar Fields
- For Vector Fields
- Line Integral in Differential Form
- Evaluating Line Integral
- Fundamental Theorem for Line Integrals
- Applications of Line Integral
- Line Integrals of Vector Valued Functions
- Examples on Line Integral
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