Evaluating Line Integrals

We know that we can use integrals to find the area under a curve, or double integrals to find the volume under a surface. But now we are going to learn about line integrals, which allow us to find the area under a surface along a particular path within that surface. In doing this we will revisit parametric equations, so you know it'll be tons of fun!

Script by Howard Whittle

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[ProfessorDaveExplains]

Whoops! At 9:53 I accidentally put three P's instead of P, Q, R. Sorry!

You’ve cleared an entire semesters worth of work up for me in 10 minutes. Thank you so much!

[femboy]

I absolutely love your videos. I know you're struggling with viewership but please don't leave youtube, you're an amazing YouTuber and are of so much help! You're the reason I've been able to learn calculus, vectors, trigonometry and everything else in 9th grade. I love you man!

[asmellygoose]

You might not see this Professor Dave, but I literally spent days trying to understand this topic, literally almost no video could help explain it for a person like me. You explained the theory and how to apply this knowledge with such simplicity. This just made my day! Thank you so much Professor dave. You are amazing!

[latifmuhammad]

I've also got 3/2 for the second Integral

[ordinaryrat]

0:29 This literally saved me. I was so confused over what the line integral was because every other person uses mathematical terms to describe it. Thanks so much!

[shivakarthik]

A lot of it covered in such a short time. The best series for Multivariable calculus.

[orjihvy]

The first part of video really helped with the intuition. It’s the clearest explanation yet on what line integral is.

[quamos]

The second comprehension questions answer should be 3/2

[kka]

Professor Dave, my daughter learned so much from you and I am forever grateful. Just a correction here, the curve C is, by definition, the curve on the x-y plane, NOT on the surface defined by f(x, y). For each small segment dS on the curve C (which is on the x-y plane), we pick a point (x, y), then evaluate f(x, y), and multiply it by dS. We then sum them all up. BTW, I have seen the error in other videos too, e.g., YouTube video taught by Michel Van Biezen. Last, I wish I could watch videos like yours when I was growing up.

[seanm]

I found myself, due to unforeseen circumstances, teaching an AP Calculus AB class, and liked doing it so much I decided to take Calculus III. I'm not sure if I could get through vector calculus without your help! Thanks!

[erockromulan]

Hey, when you are doing U-sub and writing out the integral, the notation for the lower and upper bounds should be changed to reflect what U would be when t=0, t=6. Pretty sure it should be the integral of u^1/2 from 0.25 to 144.25. It looks like you were implying that it will get changed back to the t variable, I just thought I would mention this for students watching this video. I remember getting marked down from some of my professors over this.

[jackd]

This was really really helpful. My university notes introduced it way way more complicated than this. Now that I've seen this I feel like I can now move onto the more complicated stuff that's in my uni notes, thanks!

[Wheelie]

Best explanation on youtube hands down. Thank you for helping us all pass ❤️

[sal]

Thank you Prof Dave! You not only explain the how but the what and why which are important.Now as I do the calculations it's not just numbers infront of me but I know what it is and why I'm doing what I'm doing.

[Kiky_MedPhysicist]

Thank you sir for your dedication and for making this free! 🙏

[opiret]

This has given me so much insight into how this works and was really easy to follow

[daltonmccall]

Hey professor, quick question on comprehension problem two: I get 3/2, not 33/20. I have asked some of my peers about the problem too and they get the same answer as myself. I am not sure what I am doing wrong.

Is there a walkthrough of the integral somewhere? any help would be appreciated

[orfeassiozos]

It's 1:45am. I have a lecture in less than 8 hours where we'll be doing examples on line integrals. I missed the lecture on their theory. So here I am. Thank you professor Dave.

[randomideas]

You have answered every question which was in my mind