Evaluate $integral subscript 1 superscript 3 left parenthesis straight x squared plus straight x right parenthesis space dx$ as the limit of sums.

Solution

Comparing $integral subscript 1 superscript 3 left parenthesis straight x squared plus straight x right parenthesis space dx space space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space we space get$
   $straight f left parenthesis straight x right parenthesis space equals space straight x squared plus straight x comma space space straight a space equals space 1 comma space space space straight b space equals space 3$
$therefore$    $straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 1 plus 1$
$straight f left parenthesis straight a plus straight h right parenthesis space space equals straight f left parenthesis 1 plus straight h right parenthesis space equals space left parenthesis 1 plus straight h right parenthesis squared plus left parenthesis 1 plus straight h right parenthesis straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 1 plus 2 straight h right parenthesis space equals left parenthesis 1 plus 2 straight h right parenthesis squared plus left parenthesis 1 plus 2 straight h right parenthesis$
...........................................................................
   $straight f left parenthesis straight a plus stack straight n minus 1 with bar on top right parenthesis space equals space straight f left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h squared right parenthesis space plus space left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis$
Now, $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis right square bracket$
$therefore$    $integral subscript 1 superscript 3 left parenthesis straight x squared plus straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket space left curly bracket 1 plus 1 right curly bracket space plus space left curly bracket left parenthesis 1 plus straight h right parenthesis squared plus left parenthesis 1 plus straight h right parenthesis right curly bracket space plus space left curly bracket left parenthesis 1 plus 2 straight h right parenthesis squared plus left parenthesis 1 plus 2 straight h right parenthesis right curly bracket plus...$
$.... plus left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket straight n plus 2 straight h space left curly bracket 1 plus 2 plus 3 plus..... plus stack straight n minus 1 with bar on top space right curly bracket space plus space straight h squared space left curly bracket 1 squared plus 2 squared plus 3 squared plus.......$
   $plus left parenthesis straight n minus 1 right parenthesis squared right curly bracket plus straight n plus straight h space left curly bracket 1 plus 2 plus 3 plus..... plus stack straight n minus 1 with bar on top right curly bracket right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets straight n plus 2 straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space straight n over denominator 2 end fraction plus straight h squared fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction plus straight n plus straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction close square brackets equals space Lt with straight h rightwards arrow 0 below open square brackets straight n space straight h space plus space left parenthesis straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space plus space fraction numerator left parenthesis straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space left parenthesis 2 space straight n space straight h space minus straight h right parenthesis over denominator 6 end fraction plus space straight n space straight h plus fraction numerator left parenthesis nh space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis over denominator 2 end fraction close square brackets$
$equals space Lt with straight h rightwards arrow 0 below space open square brackets 2 plus left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space plus space fraction numerator left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space left parenthesis 4 minus straight h right parenthesis over denominator 6 end fraction plus 2 plus fraction numerator left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis over denominator 2 end fraction close square brackets$
$space open square brackets because space space straight n space straight h space space equals space straight b space minus straight a space equals space 3 space minus space 1 space equals space 2 close square brackets$
   $equals 2 plus left parenthesis 2 minus 0 right parenthesis left parenthesis 2 right parenthesis plus fraction numerator left parenthesis 2 minus 0 right parenthesis left parenthesis 2 right parenthesis left parenthesis 4 minus 0 right parenthesis over denominator 6 end fraction plus 2 plus fraction numerator left parenthesis 2 minus 0 right parenthesis left parenthesis 2 right parenthesis over denominator 2 end fraction equals space 2 plus 4 plus 8 over 3 plus 2 plus 2 space equals space 38 over 3$

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Integrals

Evaluate $integral subscript 1 superscript 3 left parenthesis straight x squared plus straight x right parenthesis space dx$ as the limit of sums.

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