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

Solution

Comparing $integral subscript 0 superscript 2 left parenthesis straight x squared plus straight x plus 1 right parenthesis space dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space$ we get
   $straight f left parenthesis straight x right parenthesis space equals space straight x squared plus straight x plus 1 comma space space straight a space equals space 0 comma space space space straight b space equals space 2 straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 0 right parenthesis space equals space 0 plus 0 plus 1 space equals space 1 straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis straight h right parenthesis space equals space straight h squared plus straight h plus 1 space equals space 1 straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 2 straight h right parenthesis space equals space 2 squared straight h squared plus 2 straight h plus 1$
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$straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight f left parenthesis stack straight n minus 1 with bar on top straight h right parenthesis space equals space left parenthesis straight n minus 1 right parenthesis squared space straight h squared space plus space left parenthesis straight n minus 1 right parenthesis space straight h space plus 1$
Now,    $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h left right arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus.... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis right square bracket$
$therefore$    $integral subscript 0 superscript 2 space left parenthesis straight x squared plus straight x plus 1 right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket 1 plus left parenthesis straight h squared plus straight h plus 1 right parenthesis plus left parenthesis 2 squared straight h squared plus 2 straight h plus 1 right parenthesis plus......$
$plus open curly brackets left parenthesis straight n minus 1 right parenthesis squared space straight h squared space plus space left parenthesis straight n minus 1 right parenthesis space straight h space plus 1 close curly brackets right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket straight n plus straight h space left parenthesis 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right parenthesis space plus space straight h squared space open curly brackets 1 squared plus 2 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared close curly brackets right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets straight n plus straight h fraction numerator left parenthesis straight n minus 1 right parenthesis thin space left parenthesis straight n right parenthesis over denominator 2 end fraction plus straight h squared fraction numerator left parenthesis straight n minus 1 right parenthesis left parenthesis straight n right parenthesis left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction close square brackets equals space Lt with straight h rightwards arrow 0 below open square brackets straight n space straight h space plus space fraction numerator left parenthesis straight n space straight h right parenthesis space left parenthesis straight n space straight h space minus space straight h right parenthesis over denominator 2 end fraction plus fraction numerator left parenthesis space straight n space straight h right parenthesis space left parenthesis straight n space straight h space minus space straight h right parenthesis space left parenthesis 2 space straight n space straight h space minus space straight h right parenthesis over denominator 6 end fraction close square brackets equals space Lt with straight h rightwards arrow 0 below space open square brackets 2 plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus straight h right parenthesis over denominator 2 end fraction plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus straight h right parenthesis space left parenthesis 4 minus straight h right parenthesis over denominator 6 end fraction close square brackets$
$open square brackets because space straight n space straight h space equals space straight b space minus straight a space equals space 2 space minus space 0 space equals space 2 close square brackets$
$equals space 2 plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus 0 right parenthesis over denominator 2 end fraction plus fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis 2 minus 0 right parenthesis space left parenthesis 4 minus 0 right parenthesis over denominator 6 end fraction equals space 2 plus 2 plus 8 over 3 space equals space fraction numerator 6 plus 6 plus 8 over denominator 3 end fraction space equals space 20 over 3$

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Integrals

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

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