Line Integral Homework: How to Solve for the Total Length of a Curve

[chetzread]

Homework Statement

i'm not sure what is line integral...

Homework Equations

The Attempt at a Solution

Does it mean total length of line under the curve?

 

[SteamKing]

Homework Statement

i'm not sure what is line integral...

Homework Equations

The Attempt at a Solution

Does it mean total length of line under the curve?

If f(x,y,z) = 1, then the line integral of this function over the curve C will give the arc length between a and b. If f(x,y,z) is some other function, this won't be the case.

 

[chetzread]

If f(x,y,z) = 1, then the line integral of this function over the curve C will give the arc length between a and b. If f(x,y,z) is some other function, this won't be the case.

If f(x,y,z) is some other function , what would it be?

 

[SteamKing]

If f(x,y,z) is some other function , what would it be?

Who knows?

Not every line integral is imbued with deep physical meaning.

 

[LCKurtz]

If for example $f(x,y,z)$ is the density per unit length of a wire, say in kg/m, then the integral $\int_C f(x,y,z)~ds$ would represent the total kg for the wire.
c

 

[chetzread]

If for example $f(x,y,z)$ is the density per unit length of a wire, say in kg/m, then the integral $\int_C f(x,y,z)~ds$ would represent the total kg for the wire.
c

Then, how about s( in the first photo in first post) only ? It represents the total length of curve?

 

[Chestermiller]

Then, how about s( in the first photo in first post) only ? It represents the total length of curve?

No. s is the cumulative distance along the curve, starting from a specified location.

What they are doing here is defining the locus of points along a curve in space by specifying each of the three coordinates x, y, and z as a continuous parametric function of a parameter t.
x = x(t)
y = y(t)
z = z (t)
Specifying a value for t determines the coordinates of any particular point along the curve.

Consider two closely neighboring points along the curve that are situated at t and at t + dt. The differences in the x, y, and z coordinates of these two points are given by:
$dx = x(t+dt)-x(t)=\frac{dx}{dt}dt$
$dy = y(t+dt)-y(t)=\frac{dy}{dt}dt$
$dz = z(t+dt)-z(t)=\frac{dz}{dt}dt$
The spatial distance ds between the two neighboring points along the curve is given by the Pythagorean Theorem:
$$(ds)^2=(dx)^2+(dy)^2+(dz)^2=\left[\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2+\left(\frac{dz}{dt}\right)^2\right](dt^2)$$
If you want to find the contour integral of a function f(x,y,z) between two points along the space curve, you want to be integrating f ds. This is the same as$$\int_{t_1}^{t_2}{f\left(x(t),y(t),z(t)\right)\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2+\left(\frac{dz}{dt}\right)^2}dt}$$