text: # 12. Applications of multidimensional integrals
  
  ### Integration with change of variables
  
  If is a diffeomorphism, the following theorem holds
  
  ### Polar coordinates
  
  The most common use of the above method are polar coordinates which are defined by the diffeomorphism . Thus, .
  Moreover,
  
  so
  
  E.g. let us calculate
  
  We get:
  
  ### Volume
  
  If we want to calculate the volume of a set we can do it using one of the following three methods:
  
  where is the basis is the height at and is the area of the section for given .
  
  Assume that we would like to calculate the volume of a pyramid with a base in form of a isosceles right triangle (with boundary of axes and line ) and height , so in point the height is given by the following formula . Therefore the section at has area , so the volume is:
  
  which is consistent with our knowledge about geometry.
  
  ### Length of a curve
  
  We can also calculate a length of a curve using integrals. Given a parametrization of a curve , its length between and , is given by the following formula:
  
  Let us check this formula for the line between and . Such a line has parametrization . Therefore its length is:
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