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

Solution

Comparing $integral subscript 1 superscript 2 left parenthesis straight x squared plus straight x right parenthesis space dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space space we space get$
$straight f left parenthesis straight x right parenthesis space equals space straight x squared plus straight x comma space space straight a space equals space 1 comma space space straight b space equals space 2$
$therefore$ f(a) = f(1) = 1+1
   $straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 1 plus straight h right parenthesis space equals space left parenthesis 1 plus straight h right parenthesis squared plus left parenthesis 1 plus straight h right parenthesis straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 1 plus 2 straight h right parenthesis space equals space left parenthesis 1 plus 2 straight h right parenthesis squared plus left parenthesis 1 plus 2 straight h right parenthesis$
................................................................................
   $straight f left parenthesis straight a plus stack straight n minus 1 with bar on top right parenthesis space equals space straight f left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared space plus space left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis$
Now $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus.... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$therefore space space space integral subscript 1 superscript 2 left parenthesis straight x squared plus straight x right parenthesis dx space equals space Lt with straight h rightwards arrow 0 below left square bracket left curly bracket 1 plus 1 right curly bracket space plus space left curly bracket left parenthesis 1 plus straight h right parenthesis right curly bracket space plus left curly bracket left parenthesis 1 plus 2 straight h right parenthesis squared plus left parenthesis 1 plus 2 straight h right parenthesis right curly bracket plus.....$
   $..... plus left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets straight n plus 2 straight h fraction numerator left parenthesis straight n minus 1 right parenthesis straight n over denominator 2 end fraction plus straight h squared fraction numerator left parenthesis straight n minus 1 right parenthesis thin space left parenthesis straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction plus straight n plus straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction close square brackets equals space Lt with straight h rightwards arrow 0 below space open square brackets straight n space straight h plus left parenthesis straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space plus space fraction numerator left parenthesis straight n space straight h minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space left parenthesis 2 space straight n space straight h minus straight h right parenthesis over denominator 6 end fraction plus space straight n space straight h space plus space fraction numerator left parenthesis space straight n space straight h space minus space straight h right parenthesis space left parenthesis straight n space straight h right parenthesis over denominator 2 end fraction close square brackets$
$equals space Lt with straight h rightwards arrow 0 below open square brackets 1 plus left parenthesis 1 minus straight h right parenthesis left parenthesis 1 right parenthesis plus fraction numerator left parenthesis 1 minus straight h right parenthesis left parenthesis 1 right parenthesis left parenthesis 2 minus straight h right parenthesis over denominator 6 end fraction plus 1 plus fraction numerator left parenthesis 1 minus straight h right parenthesis left parenthesis 1 right parenthesis over denominator 2 end fraction close square brackets$
$open square brackets because space straight n space straight h space equals space straight b minus straight a space equals space 2 minus 1 space equals space 1 close square brackets$
$equals 1 plus left parenthesis 1 minus 0 right parenthesis space left parenthesis 1 right parenthesis space plus space fraction numerator left parenthesis 1 minus 0 right parenthesis space left parenthesis 1 right parenthesis thin space left parenthesis 2 minus 0 right parenthesis over denominator 6 end fraction plus 1 plus fraction numerator left parenthesis 1 minus 0 right parenthesis space left parenthesis 1 right parenthesis over denominator 2 end fraction equals space 1 plus 1 plus 1 third plus 1 plus 1 half space equals 23 over 3$

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Integrals

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

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