Evaluate $integral subscript 0 superscript 2 straight e to the power of straight x space dx$ as the limit of a sum.

Solution

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

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Integrals

Evaluate $integral subscript 0 superscript 2 straight e to the power of straight x space dx$ as the limit of a sum.

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