Line Integrals - Evaluating a Line Integral

Understanding Line Integrals: Step-by-Step Example & Formula Breakdown
In this video, I explain the concept of line integrals, provide the fundamental formula, and solve a step-by-step example. You'll see how to evaluate a line integral over a curve 𝐶, gaining an understanding of how to use parameterizations and integrate functions over paths.

What You Will Learn:

The definition and significance of line integrals in vector calculus.
The fundamental formula for calculating line integrals over the Integral of C of f(x,y)ds.
How to parametrize a curve for line integration.
A complete example solving the line integral of xy^4 ds, where the path C is the right half of a semicircle of radius 4.

In this video, you'll learn the key steps to calculate line integrals, including parameterization and setting up the integral with respect to t. By the end of this example, you'll be ready to solve line integrals on your own, enhancing your understanding of multivariable calculus and its applications to curves in the plane.

If you found this video helpful, don’t forget to subscribe, leave a comment, like the video, and share it with friends or anyone learning calculus!

Support my work and more math content on Patreon:

#LineIntegrals #Calculus #MultivariableCalculus #VectorCalculus #CalculusTutorial #PatrickJMT #CalculusExamples #LineIntegralExample #CalculusHelp #CalculusConcepts #EvaluatingLineIntegrals #PathIntegrals #ParametricCurves #CalculusProblemSolving #LearnCalculus #IntegrationOverCurves #AdvancedCalculus #MathematicalIntegration #LineIntegralExplained #CalculusIn3D #LineIntegralsInCalculus #Parametrization #CalculusTechniques

[patrickjmt]

thanks for the nice words : )

i am glad that i can help you!

[LazyChristy]

I can't believe your videos are from around 10 years ago. They're amazing and are better than most lectures/videos I've been to or seen. You're awesome. :D

[TurkeyShaman]

This exact example was assigned in my homework too.

[yousifmousa]

your explanations are better than 13 profs and 20 high school teachers combined..much appreciated!

[cmcnic]

You are fantastic! I got sick and missed most of this in class, and I was having a difficult time learning it on my own... Your videos have been so clear and easy to understand, thank you!

[patrickjmt]

@JBS90 glad you like the vids. do not worry though, when i taught i am sure plenty of people said i sucked.
yes, i am using stewarts book (my old version 5 edition that i taught from)

[justsomeoneonline]

Patrick, I know you will probably never see this, but you honestly helped me so much in my studies. With your help, I managed to do extremely well at my community college, and I transferred to UC Berkeley! I still use your videos at Cal, and I'm absolutely amazed by how you explain such complex stuff so simply!

Thank you so

[brandonhughes]

I was 5 years old when this video was released and I’m now using it to help me in physics degree. Pretty incredible that 15 years after it was released this is still one of the best videos on YouTube for learning line ints

[sreajan]

I watched a dozen of videos to get a better visulization of line integral concept. I can say... It was this video which finally made the visualization most clear.

[mathematicandlogicexplorer]

I send my gratitude to you, Patrick. Line Integrals are the hardest concept I've met.

[DeluxeWarPlaya]

This video helped me understand what a line integral actually is. That bit at the beginning about integrating the function around the curve C really helped. My maths professor didn't explain that bit. So in comparison to a regular integral, integrating about the x axis, instead of finding the area between the function and the x axis, we're finding the area between a function and some other line.
IT ALL MAKES SENSE NOW!!!

[DavidBowes]

Hey Patrick. Bro, i dont know how else to thank you for the service you provide. You got me through calc 1, 2, and 3. Im sure you get this alot, but im not sure I could have passed without your vids. Ill be majoring in mathematics at USF and I feel prepared for Diff Eq. Thanks bro. Where can I donate?

[patrickjmt]

perhaps they can just play my videos in all calc classes : )

[PhatPhunkyPhatso]

I'm so glad I found your channel dude, without your videos Electronic Engineering would've been a living hell, but you made it a breeze. Cheers!

[lilmayn]

it took my professor two classes to explain this and you just did it in less than 10 min. thank you!

[sfcsarah]

Thanks for making this video, you make this otherwise complicated math lesson very short and easy.

Btw, your use of erasing and rewriting different parts of the evaluation is very effective.

Thanks again

[huldaimeldapokolo]

Thank you sooo much Patrick, there was no why I could watch this,  have understood it in less then 10 min and not leave a comment..
you are a life saver.
thanks

[ajmiller]

Thanks Man We've been going over this for 2 days in Multi-var and I had no conceptual idea what was going on, and in 1 minute you mad it make sense, PatricJMT to save the day again!!! much appreciated bro

[zeedee]

These are awesome! You have a very clear, concise way of explaining things. Thanks for taking the time to make these.

[DavidBowes]

the boundaries for t are given, notice the integral is over the right side of a circle, thus T is between -pi/2 and +pi/2. Hope that helped.