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

Solution

Comparing $integral subscript straight a superscript straight b straight x squared dx$ with $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space space we space get comma$
   $straight f left parenthesis straight x right parenthesis space equals space straight x squared$
$therefore$ $straight f left parenthesis straight a right parenthesis space equals space straight a squared$
$straight f left parenthesis straight a plus straight h right parenthesis space equals space left parenthesis straight a plus straight h right parenthesis squared straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space left parenthesis straight a plus 2 straight h right parenthesis squared$
.............................
$straight f left parenthesis straight a plus stack straight n minus 1 with bar on top minus straight h right parenthesis space equals space left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis squared$
Now    $integral subscript straight a superscript straight b space straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis right square bracket$
where $nh space equals space straight b minus straight a$
$therefore$    $integral subscript straight a superscript straight b straight x squared dx space equals space Lt with straight h rightwards arrow 0 below straight h left square bracket straight a squared plus left parenthesis straight a plus straight h right parenthesis squared plus left parenthesis straight a plus 2 straight h right parenthesis squared plus.... plus left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis squared$
$equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket left parenthesis straight a squared plus straight a squared plus straight a squared plus..... space to space straight n space term right parenthesis space plus space 2 space straight a space straight h space left curly bracket 1 plus 2 plus 3 plus... plus stack straight n minus 1 with bar on top right curly bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space$
   $plus straight h squared left curly bracket 1 squared plus 2 squared plus 3 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared right curly bracket right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h space space open square brackets na squared plus 2 ah. fraction numerator left parenthesis straight n minus 1 right parenthesis straight n over denominator 2 end fraction plus straight h squared fraction numerator left parenthesis straight n minus 1 right parenthesis left parenthesis straight n right parenthesis left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction close square brackets equals space Lt with straight h rightwards arrow 0 below open square brackets straight a squared nh plus straight a left parenthesis nh minus straight h right parenthesis left parenthesis nh right parenthesis plus fraction numerator left parenthesis nh minus straight h right parenthesis left parenthesis nh right parenthesis left parenthesis 2 nh minus straight h right parenthesis over denominator 6 end fraction close square brackets$
$equals space Lt with straight h rightwards arrow 0 below open square brackets straight a squared left parenthesis straight b minus straight a right parenthesis plus straight a left parenthesis straight a minus straight b minus straight h right parenthesis space left parenthesis straight b minus straight a right parenthesis plus fraction numerator left parenthesis straight b minus straight a minus straight h right parenthesis space left parenthesis straight b minus straight a right parenthesis left parenthesis 2 straight b minus 2 straight a minus straight h right parenthesis over denominator 6 end fraction close square brackets equals space straight a squared left parenthesis straight b minus straight a right parenthesis plus straight a left parenthesis straight b minus straight a minus 0 right parenthesis left parenthesis straight b minus straight a right parenthesis plus fraction numerator left parenthesis straight b minus straight a minus 0 right parenthesis left parenthesis straight b minus straight a right parenthesis left parenthesis 2 straight b minus 2 straight a minus 0 right parenthesis over denominator 6 end fraction equals space straight a squared left parenthesis straight b minus straight a right parenthesis plus straight a left parenthesis straight b minus straight a right parenthesis squared plus 1 third left parenthesis straight b minus straight a right parenthesis cubed space equals space left parenthesis straight b minus straight a right parenthesis open square brackets straight a squared plus straight a left parenthesis straight b minus straight a right parenthesis plus 1 third left parenthesis straight b minus straight a right parenthesis squared close square brackets$
$equals space left parenthesis straight b minus straight a right parenthesis open square brackets fraction numerator 3 straight a squared plus 3 straight a left parenthesis straight b minus straight a right parenthesis plus left parenthesis straight b minus straight a right parenthesis squared over denominator 3 end fraction close square brackets$
$equals space fraction numerator left parenthesis straight b minus straight a right parenthesis over denominator 3 end fraction left square bracket 3 space straight a squared plus 3 space straight a space straight b space minus space 3 space straight a squared space plus space straight b squared space plus space straight a squared space minus space 2 space straight a space straight b right square bracket equals space 1 third left parenthesis straight b minus straight a right parenthesis space left parenthesis straight a squared plus straight b squared plus ab right parenthesis space equals space 1 third left parenthesis straight b cubed minus straight a cubed right parenthesis$

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Integrals

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

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