The FTOC states that if f is continuous on [ a , b ] then it is integrable.If f is not defined at certain points of [ a , b ] we can often give meaning to an improper integral. But under what circumstances will f always be integrable on it's domain, properly or improperly?If we partition [ a , b ] and consider f on the subintervals ( x i , x i + 1 ) and assume it is continuous, (so f is piecewise continuous on [ a , b ]), is f then integrable on the entire interval simply by the adding up each integral over each subinterval? But the interval is open, so how does FTOC and improper integrals come into play? Will the improper integral over [ x i , x i − 1 ] always be well-defined?

Question

The FTOC states that if   f is continuous on   [  a  ,  b  ] then it is integrable.If   f is not defined at certain points of   [  a  ,  b  ] we can often give meaning to an improper integral. But under what circumstances will   f always be integrable on it&#039;s domain, properly or improperly?If we partition   [  a  ,  b  ] and consider   f on the subintervals   (      x    i    ,      x          i      +      1        ) and assume it is continuous, (so   f is piecewise continuous on   [  a  ,  b  ]), is   f then integrable on the entire interval simply by the adding up each integral over each subinterval? But the interval is open, so how does FTOC and improper integrals come into play? Will the improper integral over   [      x    i    ,      x          i      −      1        ] always be well-defined?

Ellie Meyers · Accepted Answer

Use the definition of improper integral on each open interval where   f is continuous. For example, if   f is continuous in   [      x    i    ,      x          i      +      1        ), but not continuous on the end point       x          i      +      1      , then      ∫                  x        i                            x                  i          +          1                      f    d  x  =      lim          t      →              x                  i          +          1                −                  ∫                  x        i                    t        f    d  xhe FTOC then can be used on the interval   [      x    i    ,  t  ]. It is not necessarily always exist. If it is divergent, then the whole integral is divergent.

The FTOC states that if f is continuous on [ a , b ] then it is integrable.

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