Evaluate $integral subscript 1 superscript 3 space left parenthesis 3 straight x squared plus 2 straight x right parenthesis space dx$ as the limit of sums.

Solution

Comparing $integral subscript 1 superscript 3 space left parenthesis 3 straight x squared plus 2 straight x right parenthesis space dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space space we space get comma$
$straight f left parenthesis straight x right parenthesis space equals space 3 straight x squared plus 2 straight x comma space space straight a space equals space 1 comma space space straight b space equals space 3 straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 3 plus 2 straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 1 plus straight h right parenthesis space equals space 3 left parenthesis 1 plus straight h right parenthesis squared plus 2 left parenthesis 1 plus straight h right parenthesis straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 1 plus 2 straight h right parenthesis space equals space 3 left parenthesis 1 plus 2 straight h right parenthesis squared plus 2 left parenthesis 1 plus 2 straight h right parenthesis$
.......................................................................
   $straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space straight f left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space 3 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus 2 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis$
Now,    $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets straight f left parenthesis straight a right parenthesis space plus space straight f left parenthesis straight a plus straight h right parenthesis space plus space straight f left parenthesis straight a plus 2 straight h right parenthesis plus..... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top right parenthesis space straight h close square brackets$
   $equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket space open curly brackets 3 plus 2 right curly bracket plus left curly bracket 3 space left parenthesis 1 plus straight h right parenthesis squared plus 2 space left parenthesis 1 plus straight h right parenthesis right curly bracket plus left curly bracket 3 space left parenthesis 1 plus 2 straight h right parenthesis squared plus 2 space left parenthesis 1 plus 2 straight h right parenthesis close curly brackets space$
   $plus.... plus space left curly bracket 3 space left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared space plus space 2 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis right curly bracket right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space left square bracket 3 straight n plus 2 straight n plus 6 straight h space left curly bracket 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right curly bracket space plus space 2 straight h space left curly bracket 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right curly bracket space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space plus space 3 space straight h squared space open curly brackets 1 squared plus 2 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared close curly brackets right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets 5 space straight n plus 6 space straight h space fraction numerator left parenthesis straight n minus 1 right parenthesis space straight n over denominator 2 end fraction plus 2 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction plus 3 space straight h squared space fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction close square brackets$
   $equals space Lt with straight h rightwards arrow 0 below space open square brackets 5 space straight n space straight h space plus space 3 space left parenthesis straight n space straight h space minus space straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space plus space left parenthesis straight n space straight h space minus space straight h right parenthesis space left parenthesis space straight n space straight h right parenthesis plus space 1 half left parenthesis straight n space straight h minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space left parenthesis 2 space straight n space straight h space minus space straight h right parenthesis close square brackets$
$equals space Lt with straight h rightwards arrow 0 below open square brackets 5 cross times 2 plus 3 space left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space plus left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space plus space 1 half left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space left parenthesis 4 minus straight h right parenthesis close square brackets$
   $open square brackets because space straight n space straight h space equals space straight b space minus straight a space equals space 3 space minus space 1 space equals space 2 close square brackets$
$equals space 10 plus 6 space left parenthesis 2 minus 0 right parenthesis space plus space 2 left parenthesis 2 minus 0 right parenthesis space plus left parenthesis 2 minus 0 right parenthesis space left parenthesis 4 minus 0 right parenthesis equals space 10 plus 12 plus 4 plus 8 space equals space 34$

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Integrals

Evaluate $integral subscript 1 superscript 3 space left parenthesis 3 straight x squared plus 2 straight x right parenthesis space dx$ as the limit of sums.

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