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20.
Evaluate the following integrals as the limit of a sum
$integral subscript 0 superscript 2 left parenthesis straight x squared plus 1 right parenthesis space dx$

Comparing $integral subscript 0 superscript 2 left parenthesis straight x squared plus 1 right parenthesis space dx space space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space we space get$
$straight f left parenthesis straight x right parenthesis space equals straight x squared plus 1 semicolon space space space straight a space equals space 0 comma space space straight b equals space 2$
$therefore$    $straight f left parenthesis straight a right parenthesis space equals space straight a squared plus 1 comma space space straight f left parenthesis straight a plus straight h right parenthesis space equals space left parenthesis straight a plus straight h right parenthesis squared plus 1 comma space space straight f left parenthesis straight a plus 2 straight h right parenthesis space space equals left parenthesis straight a plus 2 straight h right parenthesis squared plus 1 comma$
.......,    $straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space left square bracket straight a plus left parenthesis straight n minus 1 right parenthesis space straight h squared right square bracket space plus space 1$
Now,    $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus.... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$rightwards double arrow$    $integral subscript straight a superscript straight b left parenthesis straight x squared plus 1 right parenthesis dx space equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket left curly bracket straight a squared plus 1 right curly bracket space plus space left curly bracket left parenthesis straight a plus straight h right parenthesis squared plus 1 right curly bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space$
$plus open curly brackets left parenthesis straight a plus 2 straight h right parenthesis squared plus 1 close curly brackets plus.... plus open curly brackets left parenthesis straight a plus stack straight n minus 1 with bar on top straight h squared right parenthesis plus 1 close curly brackets right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets left parenthesis straight a squared plus straight a squared plus straight a squared plus..... space to space straight n space terms right parenthesis plus space open curly brackets 1 plus 1 plus 1 plus.... plus space to space straight n space terms close curly brackets close square brackets space space space plus space straight h squared space open curly brackets 1 squared plus 2 cubed plus 3 squared plus..... plus left parenthesis straight n minus 1 right parenthesis squared close curly brackets space plus space 2 ah open curly brackets 1 plus 2 plus 3 plus.... plus left parenthesis straight n minus 1 right parenthesis close curly brackets right square bracket$

$equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets straight n space straight a squared plus straight n plus straight h squared fraction numerator left parenthesis straight n minus 1 right parenthesis space straight n space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction plus 2 space straight a space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space straight n over denominator 2 end fraction close square brackets$

$equals space Lt with straight h rightwards arrow 0 below open square brackets straight a squared left parenthesis nh right parenthesis space plus nh plus fraction numerator left parenthesis nh minus straight h right parenthesis thin space left parenthesis nh right parenthesis space left parenthesis 2 nh minus straight h right parenthesis over denominator 6 end fraction plus straight a left parenthesis nh right parenthesis space left parenthesis nh minus straight h right parenthesis close square brackets space equals stack space Lt with straight h rightwards arrow 0 below open square brackets 0 space left parenthesis 2 right parenthesis plus 2 space plus space fraction numerator left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space left parenthesis 4 minus straight h right parenthesis over denominator 6 end fraction plus left parenthesis 0 right parenthesis space left parenthesis 2 right parenthesis space left parenthesis 2 minus straight h right parenthesis close square brackets$
$left square bracket because space nh space equals space straight b minus straight a space equals space 2 minus 0 space equals space 2 right square bracket$
   $equals 0 plus 2 plus fraction numerator left parenthesis 2 minus 0 right parenthesis space left parenthesis 2 right parenthesis space left parenthesis 4 minus 0 right parenthesis over denominator 6 end fraction space equals 0 space equals space 2 plus 8 over 3 space equals space 14 over 3$