Evaluate
$integral subscript straight a superscript straight b space straight e to the power of straight x space dx$ as limit of sum.

Solution

Comparing $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space with space integral subscript straight a superscript straight b space straight e to the power of straight x dx comma space space we space get comma$
   $straight f left parenthesis straight x right parenthesis space equals space straight e to the power of straight x$
$therefore$ $straight f left parenthesis straight a right parenthesis space equals space straight e to the power of straight a comma space space straight f left parenthesis straight a plus straight h right parenthesis space equals space straight e to the power of straight a plus straight h end exponent comma$
   $straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight e to the power of straight a plus 2 straight h end exponent space space...... space space straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight e to the power of straight a plus left parenthesis straight n minus 1 right parenthesis straight h end exponent$
Now,    $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$rightwards double arrow$ $integral subscript straight a superscript straight b straight e to the power of straight x dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight e to the power of straight a plus straight e to the power of straight a plus straight h end exponent plus straight e to the power of straight a plus 2 straight h end exponent plus... plus straight e to the power of straight a plus left parenthesis straight n minus 1 right parenthesis straight h end exponent right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h space straight e to the power of straight a space open square brackets fraction numerator 1 left parenthesis straight e to the power of nh minus 1 end exponent right parenthesis over denominator straight e to the power of straight h minus 1 end exponent end fraction close square brackets$ $open square brackets because space space space straight S subscript straight n space equals space straight a fraction numerator left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction close square brackets$
   $equals space Lt with straight h rightwards arrow 0 below space straight h space straight e to the power of straight a space open square brackets fraction numerator straight e to the power of straight b minus straight a end exponent minus 1 over denominator straight e to the power of straight h minus 1 end fraction close square brackets space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space space nh space equals space straight b minus straight a right square bracket$
   $equals space straight e to the power of straight a space left parenthesis straight e to the power of straight b minus straight a end exponent minus 1 right parenthesis space Lt with straight h rightwards arrow 0 below space open parentheses fraction numerator 1 over denominator begin display style fraction numerator straight e to the power of straight h minus 1 over denominator straight h end fraction end style end fraction close parentheses$
   $equals space left parenthesis straight e to the power of straight b minus straight e to the power of straight a right parenthesis. space 1 over 1 space equals space straight e to the power of straight b space minus space straight e to the power of straight a$

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Integrals

Evaluate
$integral subscript straight a superscript straight b space straight e to the power of straight x space dx$ as limit of sum.

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