Understanding Arc Length and Line Integrals for Surface Area Calculation

[rppearso]

Hello, I am trying to solve for the surface area of a odd surface for a fire relief PSV and needed to do a line integral but I was reading into my calculus book and going back to the definition of arc length I am confused:

L = lim_{n\rightarrow\infty}\sum (P_{i-1}*P_{i})

Multiplication should give you an area like when you multipy the function of a curve C by the length just like an integral. This is a fundamental definition so I can not go back any farther maybe I am just missing something logically in my own mind. Understanding line integration is also fundamental to fluid dynamics and E&M which are both of great interest to me.

 

[HallsofIvy]

Hello, I am trying to solve for the surface area of a odd surface for a fire relief PSV and needed to do a line integral but I was reading into my calculus book and going back to the definition of arc length I am confused:

L = lim_{n\rightarrow\infty}\sum (P_{i-1}*P_{i})

First what are P_i and P_i-1? Without knowing that we can't make any sense out of that formula. Also, where did you get that formula? I don't recognize it.
If you divide a curve into n pieces, then the arclength is the sum of the length of anyone piece which, for large n, can be approximated by the straight line between the endpoints:
sum \sqrt{((x_i- x_{i-1})^2+ (y_i- y_{i-1})^2+ (z_i- z_{i-1})^2} and the actual arc length is the limit as n goes to infinity.

Multiplication should give you an area like when you multipy the function of a curve C by the length just like an integral.

I have no idea what you mean by this. By "the function of a curve" I presume you mean something like the f(x) in y= f(x) or the vector function \vec{r}(t)= f(t)\vec{i}+ g(t)\vec{j}+ h(t)\vec{k}. But the product of either of those by a number (the arclength) is not a number (the area). And what area are you talking about? In general, an arc does not have any "area" associated with it.

This is a fundamental definition so I can not go back any farther maybe I am just missing something logically in my own mind. Understanding line integration is also fundamental to fluid dynamics and E&M which are both of great interest to me.

In general, if a curve in 3 dimensions is given by (x(t),y(t),z(t)), that is, as parametric functions of t, then the arclength, from t₀ to t₁ is given by
\int_{t_0}^{t_1}\sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2+ \left(\frac{dz}{dt}\right)^2} dt

 

[rppearso]

First what are P_i and P_i-1? Without knowing that we can't make any sense out of that formula. Also, where did you get that formula? I don't recognize it.
If you divide a curve into n pieces, then the arclength is the sum of the length of anyone piece which, for large n, can be approximated by the straight line between the endpoints:
sum \sqrt{((x_i- x_{i-1})^2+ (y_i- y_{i-1})^2+ (z_i- z_{i-1})^2} and the actual arc length is the limit as n goes to infinity.


I have no idea what you mean by this. By "the function of a curve" I presume you mean something like the f(x) in y= f(x) or the vector function \vec{r}(t)= f(t)\vec{i}+ g(t)\vec{j}+ h(t)\vec{k}. But the product of either of those by a number (the arclength) is not a number (the area). And what area are you talking about? In general, an arc does not have any "area" associated with it.


In general, if a curve in 3 dimensions is given by (x(t),y(t),z(t)), that is, as parametric functions of t, then the arclength, from t₀ to t₁ is given by
\int_{t_0}^{t_1}\sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2+ \left(\frac{dz}{dt}\right)^2} dt

This was out of James Stewart Calculus book and I realized this just represneted a line segment it was not multiplicitive.