Evaluate $integral subscript 0 superscript 1 straight e to the power of 2 minus 3 straight x end exponent space dx$ as a limit of a sum.

Solution

Comparing $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space with space integral subscript 0 superscript 1 straight e to the power of 2 minus 3 straight x end exponent space dx comma space$ we get
   $straight f left parenthesis straight x right parenthesis space equals space straight e to the power of 2 minus 3 straight x end exponent comma space space straight a space equals space 0 comma space space space straight b space equals space 1$
$therefore$ $straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 0 right parenthesis space equals space straight e to the power of 2 minus 0 end exponent space equals space straight e squared$
   $straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis straight h right parenthesis space equals space straight e to the power of 2 minus 3 straight h end exponent straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 2 straight h right parenthesis space equals space straight e to the power of 2 minus 6 straight h end exponent$
.............................................
   $straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space straight f left parenthesis stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight e to the power of 2 minus 3 left parenthesis straight n minus 1 right parenthesis space straight h end exponent$
$integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis space plus space straight f left parenthesis straight a plus straight h right parenthesis space plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets straight e squared plus straight e to the power of 2 minus 3 straight h end exponent plus straight e to the power of 2 minus 6 straight h end exponent plus.... plus straight e to the power of 2 minus left parenthesis straight n minus 1 right parenthesis straight h end exponent close square brackets equals space Lt with straight h rightwards arrow 0 below space straight e squared straight h space open square brackets 1 plus straight e to the power of negative 3 straight h end exponent plus straight e to the power of negative 6 straight h end exponent plus.... plus straight e to the power of negative left parenthesis straight n minus 1 right parenthesis space straight h end exponent close square brackets equals space Lt with straight h rightwards arrow 0 below space straight e squared straight h open square brackets fraction numerator 1 left curly bracket 1 minus left parenthesis straight e to the power of negative 3 straight h end exponent right parenthesis to the power of straight n over denominator 1 minus straight e to the power of negative 3 straight h end exponent end fraction close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight S subscript straight n space equals space fraction numerator straight a left parenthesis 1 minus straight r to the power of straight n right parenthesis over denominator 1 minus straight r end fraction close square brackets$
$equals space Lt with straight h rightwards arrow 0 below straight e squared open square brackets fraction numerator 1 minus straight e to the power of negative 3 space straight n space straight h end exponent over denominator begin display style fraction numerator 1 minus straight e to the power of negative 3 straight h end exponent over denominator straight h end fraction end style end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below space straight e squared open square brackets fraction numerator 1 minus straight e to the power of negative 3 left parenthesis 1 right parenthesis end exponent over denominator begin display style fraction numerator straight e to the power of negative 3 straight h end exponent minus 1 over denominator negative 3 straight h end fraction end style left parenthesis 3 right parenthesis end fraction close square brackets space space space space left square bracket because space straight n space straight h space equals space 1 space minus space 0 space space equals space 1 right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight e squared space open square brackets fraction numerator 1 minus straight e to the power of negative 3 end exponent over denominator 1 cross times 3 end fraction close square brackets space space space$    $open square brackets because space space Lt with straight x rightwards arrow 0 below space straight e to the power of straight x minus 1 end exponent over straight x equals space 1 close square brackets$
$equals space 1 third straight e squared left parenthesis 1 minus straight e to the power of negative 3 end exponent right parenthesis space equals space 1 third left parenthesis straight e squared minus straight e to the power of negative 1 end exponent right parenthesis$

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Integrals

Evaluate $integral subscript 0 superscript 1 straight e to the power of 2 minus 3 straight x end exponent space dx$ as a limit of a sum.

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