1.
How would you explain the Big Idea of a line integral to someone who has only taken single-variable calculus (If you know someone like that, try it out with them!)
Section 1.2 Introduction to Line Integrals
Our first new object of study is called a line integral. In 1st year calculus, we study the integral of function over some interval \([a,b]\) such as \(f(x)=x^2\) on \([0,1]\text{.}\) The integral gave the area under the curve above that interval. But now imagine you have a height \(f(x,y)\) over top of not just a simple interval, but some curve in the 2D plane. Can we still make sense of a type of area under the function above that curve? This will be the notion of the Line Integral.
Pre-requisites: The derivation in this video is fairly analogous to the derivation for arclength from multivariable calculus (Math 200 at UVic). You can check out that video here. 1
Learning Objectives:
- Define Line Integrals
- Interpret Line Integrals Geometrically
Post-Video Activities Post-Video Activities
2.
We had two expressions for the line integral, one in terms of \(ds\) and one in terms of \(dt\text{.}\) Why? What was the difference? When might we use one or the other?
www.youtube.com/watch?v=e6dkE6T391c
