Line Integrals of Vector Valued Functions
Let, r be a vect function defined in t such that,
r(t) = x(t)i + y(t)j a ≤ t ≤ b
It r(t) is differentiable on a smooth curve C then,
∫C f(x, y) . dr = ∫ab f{x(t), y(t)}.√{(x'(t)2 + y'(t)2} dt
if, r(t) = x(t)i + y(t)j + x(t)k a ≤ t ≤ b
∫C f(x, y) . dr = ∫ab f{x(t), y(t), z(t)}.√{(x'(t)2 + y'(t)2+ z'(t)2} dt
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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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