#line integrals

[busybeestudent]

05.07.20

math time! vector fields, line integrals, and green’s theorem today. hoping that my big exam on tuesday goes okay!

[day 7/30 of productivity]

[idkifuknow]

Slide into my dm’s if you’ve mastered upto and beyond Green’s Theorem in calc 3.

[rebrobindoesmath]

Study With Me: Line Integrals

Hey guys! I’m currently studying for the Mathematics Subject Test of the GRE, which I plan on taking in the fall. One of the ways I like to study is by explaining the material to someone else. I currently have weekends off from research, and since Saturdays are for the boys, it leaves Sundays for GRE preparation. 

Because of this, every Sunday, I’ll explore a different undergraduate topic that could appear on the Mathematics Subject Test. This week: Line Integrals.

I’ll talk about the following:

What is a line integral?

How do you calculate a line integral?

An Example

As a brief note, this post contains LaTeX code and will be much easier to read when viewed directly on my blog, where the code will compile!

What is a line integral?

Let’s first recall what we already know about integrals. We’re used to integrating functions of one variable over an interval [a,b]. We can think of this as integrating over the path on the x-axis from a to b, and the value of the integral as giving the area bounded by the curve y=f(x) over the path [a,b]. 

But! We can also integrate over paths that aren’t just straight lines along the x-axis. The resulting integral is called a curve, contour, or path integral. Most commonly, it is known as a line integral. 

In this post, I’ll be talking about line integrals with respect to arc length. 

Before we get into it, I’d like to start by defining what it means for a curve to be smooth. A curve, C, with parameterization r(t) = <x(t), y(t)> is smooth if the derivative r’(t) is continuous and nonzero. Additionally, we can say C is piecewise smooth if it is composed of a finite number of smooth curves joined at consecutive endpoints. Basically, this means I have a bunch of curves $C_1, C_2, ..., C_n $ that are all individually smooth and Each $C_i$ has its endpoints connected to $C_{i-1}$ and $C_{i+1}$. 

Back to line integrals. Suppose we have a function f(x,y) and a smooth curve, C, in the x-y plane. We want to think about breaking C into n tiny pieces of arc length $\Delta s_i$. For each of the tiny pieces of C, choose any point $P_i = (x_i, y_i)$ and then multiply $f(P_i) = f(x_i, y_i)$ by the length $\Delta s_i$. This process is fairly similar to how we define integration for the case where the path is a line on the x-axis. We want to sum up these multiplied terms for all n terms. If the value of that sum approaches a finite, limiting value as $n \rightarrow \infty$, then the result is the line integral of f along C with respect to arc length. Below is a comparison of the single variable case integrating over a path [a,b] on the x-axis (left) and the line integral with respect to arc length over the curve C (right). 

Note the notation used for the line integral. If we’re integrating over a path C, we write C at the bottom of the integral. 

What does this mean geometrically?

The value of this integral is the area of the region whose base is C and whose height above each (x,y) point is given by f(x,y). 

How do we actually calculate the line integral?

First, parameterize C. That is, for a parameter t, find the equations x=x(t) and y=y(t)  for $a \leq t \leq b$. We consider C to be directed, which means we’re saying that we trace C in a definite direction, which is called the positive direction. Basically, we’re saying that t runs from a to b, so A = (x(a), y(a)) is the initial point and B = (x(b), y(b)) is the final point. 

Since we have $(ds)^2 = (dx)^2 + (dy)^2$ (think Pythagorean theorem), we can write:

$\frac{ds}{dt} = \pm \sqrt{( \frac{dx}{dt})^2+(\frac{dy}{dt})^2}$

which can then be rewritten as:

$\pm \sqrt{(x’(t))^2+(y’(t))^2}$.

We use the + sign if the parameter t increases in the positive direction on C and the - sign if t decreases in the positive direction on C. 

So, we have:

$\int_C f \,ds = \int_{a}^{b} f(x(t), y(t))\frac{ds}{dt} \,dt$.

An Example

Determine the value of the line integral of the function f(x,y) = x + y^2 over the quarter-circle x^2 + y^2 = 4 in the first quadrant, from (2,0) to (0,2).

Solution below.

First, we should parameterize the curve C. 

Our parameterization:

$x = 2\cos{t}$

$y = 2\sin{t}$

with $0 \leq t \leq \pi/2$.

Then we find $ds$. 

$ds = \pm \sqrt{(x’(t))^2+(y’(t))^2}$

$ds = \pm \sqrt{(-2\sin{t})^2 + (2\cos{t})^2}$

Using the fact that $\sin^2{t}+\cos^2{t} = 1$,

$ds = \pm \sqrt{4} = \pm 2$

We use positive 2, since t increases as we traverse C in the positive direction. 

So, we have:

$\int_C f \,ds = \int_{0}^{\pi/2}(2\cos{t}+4\sin^2{t})*2 \,dt$

I’ll leave the integration as an exercise, but the final answer is 2(2+$\pi$). 

Thanks for reading!

As always, feel free to send a message or an ask with any questions, comments, or concerns. Stay positive, friend!

[maryssnotes]

[mathwork]

[tswlft]

I hold my self to a certain rigour when it comes to understanding maths. Like most people are content with regurgitating formulas and cramming for exams. Whereas I dont claim to understand something until i really know the nooks and crannies and have a total intuitive comprehension of what's going on. And its truly 110% why i was so successful in year 12, cause I knew that syllabus like the back of my hand. There was nothing you could throw at me that I was not prepared for or have not seen. It's obvious that some people don't pursue that type of understanding, cause I was talking to some people the other day about tutoring and they don't want to tutor because they are too afraid of teaching them the wrong stuff, or having a misunderstanding that transfers to the student. Whereas I have never had the hesitation because I know I can tutor year 11 and 12 maths and physics with 100% understanding and not be afraid Im telling them the wrong thing. I strive to teach them everything in an intuitive way, and get them as comfortable with it as I am. Like you gotta just know that course inside and out if you want to be successful. You cant be getting into an exam and look at a question and having no idea where to start because you have not seen something like that, or not have the skills and understanding to tackle a problem you've never seen. They go hand in hand.

Moral of the story I have spent this weekend understanding line and surface integrals and integrals over R^3. Green's, Stokes' and Gauss' Theorems. And I I have a good intuitive foundation for these theorems now, and can relate them all to each other. I owe large part of that understand to Khan Academy. Who shares my belief in having intuitive understanding of a topic is 12/10 the most important thing. So it annoys me when I say, I dont understand something. And somewhen else is like 'oh its eassy its just this formula' and then Im like 'okay, explain it to me'. And literally 9/10 they cant. Like that is not understanding it. If you call that understanding then I understand it 12x better than you and I still claim I don't understand. I bet the majority of people in my multivariable and complex calculus have no idea what a line integral actually represents. But in saying all of this intuitively understanding something is nothing if you don't actually know the definition and can answer real questions. Its interesting though cause a postgrad was talking about this the other day and was lke 'you dont have to understand any of it, just know how to use the formulas and answer questions, it will all make sense and you'll understand it next year'. Which is true, you never understand something that well when you're being taught it, but the next year when you use those results it all sorta comes full circle and you get it. But why wait? Try your best to be at that level now and then you cant get fucked over in an exam.

Long story short, basic ass niggas, get on my level.

[ifikeyboardsmashdoesitcount]

If you give a Caitlin a calc quiz

Right now my school life is like a children's book. If you give a Caitlin a Calc quiz she'll spend 30 minutes staring at it. If she spends 30 minutes staring at it she'll want a snack. If she gets a snack she'll want some water. If she gets a glass of water she'll spill some on herself. If she spills some on herself she'll go to the bathroom to dry it off. If she goes in the bathroom she'll see her hair in the mirror. If she sees her hair in the mirror she'll realize her bangs are too long. And this the story of how I went from doing math to cutting my hair.

[nathanielbuildsatesseract]

Aaaand we've started line integrals.

Good luck, man.