Integral Calculus - Line Integral 1

Minggu, 26 Juli 2020

Line Integral 1

 

• Area of a Region R Bounded by the Curve C

    

• Path Independence of Line Integrals

  The line integral of a vector function

  is said to be path independent, if and only if P, Q and 

  R are continuous in a domain D, and if there exits some 

  scalar function u = u (x, y, z) (a scalar potential) in D 

  such that.

 

• Test for a Conservative Field

   A vector field of the formis called a 

   conservative field. The line integral of a vector 

   function is path independent if 

   and only if

      

      

   If the line is taken in xy-plane so that

      

      

   Then the test for determining if a vector field is 

   conservative can be written in the form 

      

• Length of a Curve

      

   where C is a piecewise smooth curve described by 

   the position vector, α ≤ t ≤ β.

 

  If the curve C is two-dimensional, then

     

  If the curve C is the graph of a function y = f(x) 

  in the xy-plane (α ≤ x ≤ β), then

    

• Length of a Curve in Polar Coordinates

     

  where the curve C is given by the equation r = r(θ), 

  α ≤ θ ≤ β in polar coordinates.

 

• Mass of a Wire

     

  where ρ(x, y, z) is the mass per unit length of the wire. 

  If C is a curve parametrized by the vector function

 , then the mass can be computed by 

     the formula.

   

If C is a curve in xy-plane, then the mass of the wire is given by

 

 

• Center of Mass of a Wire

      

• Moments of Inertia

  The moments of inertia about the x-axis, y-axis, 

  and z-axis are given by the formulas,

    

 

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