What is a LINE INTEGRAL? // Big Idea, Derivation & Formula

My Vector Calculus playlist: • Calculus IV: Vector Ca...
A line integral - sometimes called a path integral - is an accumulation of something along a curve (again sometimes called a path). In this first video we will really focus on a curve living in 2D, with a height above that curve, and answer the question of what is the surface area that results? This will be four first major application of line integrals. The plan for this video is to introduce the big idea of line integrals, come up with a sensible definition using Reimann integration and breaking this problem into a lot of little rectangles, and then finally deriving a formula that let's us compute this easily.
0:00 Intuitive Idea
1:39 Geometric Picture
5:12 Motivating the Definition
9:30 Deriving the Formula
11:50 Line Integral Formula
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[@ricardoalvarez4976]

12:00 hey guys just a tip, this can be also written as the integral of f(g(t),h(t)) . |dr/dt| . dt or the integral with respect to t of the function parametrized, times magnitude of the derivative of vector function with respect to t

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Geometry we can visualize this whole derivation as just pulling the curve C into a straight line, computing it’s arc-length, finding what the “pulled-tight” version of the curve above C is, laying it out on the x axis and then just performing a regular integral on that curve on an interval equal to the arc-length of C

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