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### Course: Multivariable calculus>Unit 4

Lesson 1: Line integrals for scalar functions

# Line integral example 2 (part 1)

Line integral over a closed path (part 1). Created by Sal Khan.

## Want to join the conversation?

• at the beginning of the vid. When y=0 why isn't the line along the x-axis? My intuition says it should be..
• It is along the x-axis, it's just going upwards in the z-direction making it look like it's in the x-y plane.
• I can't understand the initial line that Sal draws when y = 0 in the beginning of video, from which the parabolic surface is shown to arise. Should it not be the x axis itself. In that case we will not have the curtain between parabolic surface and x axis.
• When y = 0 then f(x,y) = x.
f(x,y) is the height of the graph along the z axis.

The first line is z=f(x,y)=x+0², or, z=x, which is a line that rises up above the xy plane at a 45 degree angle and is positioned directly over the x axis (since the x axis is where y=0). When x=0, z=0, when x=1, z=1, when x=2, z=2.
That means there is a curtain along the x axis whose height, z is given by z=x.
• I don't understand the draw, he draws a straight yellow line and after that a bunch of parabolas going up in the Z direction, a yellow circle and two lines for the travel of integration but after that¿ how is that he knows that the orange wall has that strange form?¿and how does he knows the form of the violet and yellow walls?
• Because he defined them. That is what Sal is doing in the beginning. He defines the boundries of the problem as 0<= x <=2, 0<+y<+2 and the arc from (2,0) to (0,2). That is the Line we want to integrate around ( in three pieces.)
• Correct me if I'm wrong, but if you're evaluating a closed loop line integral in a conservative vector field, then the answer will always be zero regardless of path?
• at why does Sal evaluate one of the Sin(t)'s at Pi all of a sudden (instead of Pi/2)? I got for that problem:
2Pi + 2 and sal has 4 + 2Pi. Is this a mistake?
• The second expression with a sine was sin(2t), which, evaluated at pi/2, is sin(2pi/2)=sin(pi)=0
(1 vote)
• why it can change x = 2cost ? is it defined so that x(t) = 2cost ? where this relationship is defined ?
• This is parametrization.

Here t is defined as the angle from the x axis on the quarter circle path. Always x squared plus y squared will equal 4. As on the unit circle, x and y are proportional to cos(theta) and sin(theta, respectively.

As he says in the video, it may also help to review parametrization. A single variable t is allowed to vary, and x and y move with it at the same time, tracing out a path that represents the way they are dependent on one another.

Hope this helps! I'm happy to offer more clarification if needed.
• I can never tell in this whole series if the dS and other "d"s are supposed to be the ordinary differential lower case "d" or the partial derivative symbol ∂... which are they? Does it matter very much?
• It doesn't matter when integrating since you only integrate with respect to one variable at a time anyway. You just need to be careful about it when taking derivatives since the result will differ if you do a partial or total derivative.
• At about , you mentioned that the squareroot of 4 is just 2. Isn't it important to remember that it could also be -2?
• He was looking for the distance , and distance is always positive.