Integral Calculus - Line Integral 2

Senin, 27 Juli 2020

Line Integral 2

 

• Area of a Region Bounded by a Closed Curve

    

 

Figure

 

   If the closed curve C is given in parametric 

   form, then the area can be 

   calculated by the formula,

      

• Volume of a Solid Formed by Rotating a Closed Curve about the x-axis

      

 

Figure

 

• Work

   Work done by a forceon an object moving 

   along a curve C is give by the line integral,

      

   Whereis the vector force field acting on the 

   object,is the unit tangent vector.

 

Figure

 

   If the object is moved along a curve C in the xy-plane, 

   then

  If a path C is specified by a parameter t (t often means 

  time), the formula for calculating work becomes

    

  where t goes from α to β.

 

  If a vector fieldis conservative and u(x, y, z) is a 

  scalar potential of the field, then the work on an object 

  moving from A to B can be found by the formula 

  W = u(B) – u(A).

 

• Ampere’s Law

    

  The line integral of a magnetic fieldaround a 

  closed path C is equal to the total current I flowing 

  through the area bounded by the path.

 

Figure

 

• Faraday’s Law

 

    

  The electromotive force (emf) ε induced around a 

  closed loop C is equal to the rate of the change of 

  magnetic flux ψ passing through the loop.

 

Figure

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