Another Look at Line Integrals

The arc length function s(t) of a curve r(t), t going from a to b, is defined by
s(t)=Integral from a to t of ||r'(t)||

By the Fundamental Theorem of Calculus, s'(t)=||r'(t)||

If you look at a function f(x,y,z) defined along a curve r(t) (for t from a to b) and you consider the curve as being made up of "infinitesimal" pieces of arc lengths of the curve (ds) , the work or flow along the curve (hence the line integral) becomes Integral from a to b of f ds. You often see line integral expressed this way.

If you have a parametrized curve r(t), we reconcile the two approaches in this way:
ds=(ds/dt) dt=||r'(t)|| dt

So f ds is f(x(t),y(t),z(t)) ||r'(t)|| dt

Note that Integral ds is Integral ||r'(t)|| dt which is arc length.

Flow along a curve and work are two different physical ideas but they are represented by the same thing: the line integral.

In[19]:=

  f[x_,y_]:=x^2 y

In[20]:=

  r[t_]:={3 Cos[t],3 Sin[t]}

The integral of f ds from 0 to Pi is: (let's go piece by piece)

In[21]:=

  f[r[t][[1]],r[t][[2]]]

Out[21]=

           2
  27 Cos[t]  Sin[t]

In[22]:=

  norm2[D[r[t],t]]

Out[22]=

               2           2
  Sqrt[9 Cos[t]  + 9 Sin[t] ]

In[23]:=

  Simplify[%]

Out[23]=

In[24]:=

  fds=3 f[r[t][[1]],r[t][[2]]]

Out[24]=

           2
  81 Cos[t]  Sin[t]

In[25]:=

  Integrate[81 Cos[t]^2 Sin[t],{t,0,Pi}]

Out[25]=

We can do this all at once, of course.

In[26]:=

  Integrate[f[r[t][[1]],r[t][[2]]]*norm2[D[r[t],t]],
      {t,0,Pi}]

Out[26]=

Up to Some Multivariable Calculus Ideas Part II