Evaluate $integral subscript 0 superscript 3 straight e to the power of straight x space dx$ as the limit of sums.

Solution

Comparing $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space with space integral subscript 0 superscript 3 straight e to the power of straight x dx comma space we space get$
$straight f left parenthesis straight x right parenthesis space equals space straight e to the power of straight x comma space space space straight a space equals space 0 comma space space space straight b space equals space 3 straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 0 right parenthesis space equals space straight e to the power of 0 space equals space 1 straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis straight h right parenthesis space equals space straight e to the power of straight h straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis space 2 space straight h right parenthesis space equals space straight e to the power of 2 straight h end exponent$
.....................................................
   $straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight f left parenthesis stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight e to the power of left parenthesis straight n minus 1 right parenthesis space straight h end exponent$
Now $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$rightwards double arrow$ $integral subscript 0 superscript 3 straight e to the power of straight x dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket 1 plus straight e to the power of straight h plus straight e to the power of 2 straight h end exponent plus.... plus straight e to the power of left parenthesis straight n minus 1 right parenthesis space straight h end exponent right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets fraction numerator 1 left parenthesis straight e to the power of straight n space straight h end exponent minus 1 right parenthesis over denominator straight e to the power of straight h minus 1 end fraction close square brackets$    $open square brackets because space space straight S subscript straight n space equals space fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction close square brackets$
   $equals Lt with straight h rightwards arrow 0 below space straight h open square brackets fraction numerator straight e cubed minus 1 over denominator straight e to the power of straight h minus 1 end fraction close square brackets$    $open square brackets because space nh space equals space straight b minus straight a space equals space 3 minus 0 space equals space 3 close square brackets$
   $equals left parenthesis straight e cubed minus 1 right parenthesis space Lt with straight h rightwards arrow 0 below space fraction numerator 1 over denominator begin display style fraction numerator straight e to the power of straight h minus 1 over denominator straight h end fraction end style end fraction space equals space left parenthesis straight e cubed minus 1 right parenthesis. space 1 over 1 space equals space straight e cubed minus 1$

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Integrals

Evaluate $integral subscript 0 superscript 3 straight e to the power of straight x space dx$ as the limit of sums.

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