Line integral example 2 (part 1) | Multivariable Calculus | Khan Academy

Line integral over a closed path (part 1)

Missed the previous lesson?

Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.

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[petuncha]

Its so cool how math never changes. This video could be 10 years old and still helping people in 2020 to learn calc 3.

[markolazarevic]

Thx man your explanations are very useful. The explanation in the text book I am studying this from are really confusing and incomplete. You are making this much more easier to understand. Keep going good job

[elbodi]

thank u sir i didnt understand anythin about this until i see this video and now i can progress

[saadhassan]

Sal is the best Maths teacher. He gives such a nice intuition.

[renzo]

I'm in grade 12 and somehow trying to wrap my head around this and I feel like I am almost there. It is just difficult to visualize x+y² and I am not a big fan of geometric equations, but these videos are very helpful!!!

[cookieMo]

I love the colorrrssss :D made me happy :P

[proberts]

Yes sir. It's an infinitely thin sheet from the upper bounding curve or plane to the lower bounding curve or plane. It's essentially the integrals you did in two dimensions, but the "plane" that it is traveling in can be bent into three space.

[kungman]

Thank you very much for your reply I now understand what it means

[wreynolds]

If this was given as a problem to students, it's a bit vague as to whether or not they'd be given. The difficulty of the problem depends partially on that.
In practice, here it makes no difference since this problem builds upon what we know from the last problem, where we had a similar parametrization.

[NTMihaila]

Fascinating! I've been out of math for a while, and clearly I've been missing out. This is really enjoyable. 

[bobmarleykagan]

Fascinating video than you very much. But "ds" that you showed was wrong, it must be on the 2 dimensional plane not on the third...

[ryujinryuk]

how do u get the circle with radius 2 ?

[laxgirl]

i'm still confused--are his parametrizations given? or do we need to parametrize the curves ourselves?

[theopenfamilychurch]

What would the question look like if written?

[jewbinson]

At 0.25 you say "if I graph this when y = 0" but then you draw the line y = x.

But surely when y = 0, we get f(x, 0) = x which implies that z = f(x, y) = x (I don't see how f(x) = y)

I don't see where the line y = x comes from, please can you explain?

[matteofioretti]

2024 still here. Best explanation ever.

[StiegeNZ]

Aw, I'm sad that more people haven't watched these... Ya think Sal will ever do some quantum physics stuff?

[RileyJade]

Is your integral of the curve correct? You have the integral of -4cos2t= -2sin2t, but shouldn't it be positive? There is a negative sign in the double integral formula you used, and the integral of cos is negative sin, so it should cancel and be positive. Or am I missing something?

[Ahfjo]

The outline of an object like your hand, a tree, the earth, etc.

[MrBrew]

Bad graph sal. You said when y is equal to zero as you drew the line y is equal to x... take the whole thing and rotate it so that line is the x axis.