Evaluate as limit of sums $integral subscript 1 superscript 4 left parenthesis 3 straight x squared plus 2 straight x right parenthesis space dx.$

Solution

Comparing $integral subscript 1 superscript 4 left parenthesis 3 straight x squared plus 2 straight x right parenthesis space dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space we space get$
$straight f left parenthesis straight x right parenthesis space equals space 3 straight x squared plus 2 straight x comma space space space straight a space equals space 1 comma space space straight b space equals space 4 straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 3 space plus space 2 straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 1 plus straight h right parenthesis space equals space 3 left parenthesis 1 plus straight h right parenthesis squared plus 2 left parenthesis 1 plus straight h right parenthesis straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 1 plus 2 straight h right parenthesis space equals space 3 left parenthesis 1 plus 2 straight h right parenthesis squared plus 2 left parenthesis 1 plus 2 straight h right parenthesis$
.............................................................
$straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space straight f left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space 3 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus 2 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis$
Now, $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus.... plus straight f left parenthesis straight a plus top enclose straight n minus 1 end enclose right parenthesis straight h close square brackets$
$equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket space left curly bracket 3 plus 2 right curly bracket plus left curly bracket 3 left parenthesis 1 plus straight h right parenthesis squared plus 2 left parenthesis 1 plus straight h right parenthesis right curly bracket space plus space left curly bracket 3 space left parenthesis 1 plus 2 straight h right parenthesis squared plus 2 left parenthesis 1 plus 2 straight h right parenthesis right curly bracket$
   $plus..... plus left curly bracket 3 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared space plus space 2 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis right curly bracket right square bracket$
   $equals space Lt with straight h rightwards arrow 0 below left square bracket 3 space straight n space plus space 2 space straight n space plus space 6 space straight h space left curly bracket 1 plus 2 plus..... plus stack straight n minus 1 with bar on top right curly bracket space plus space 2 straight h left curly bracket 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right curly bracket$
   $plus 3 straight h squared left curly bracket 1 squared plus 2 squared plus..... left parenthesis stack straight n minus 1 with bar on top right parenthesis squared right curly bracket right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets 5 straight n space plus 6 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis straight n over denominator 2 end fraction plus 2 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction plus 3 straight h squared fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below space open square brackets 5 space straight n space straight h space plus space 3 space left parenthesis straight n space straight h space minus space straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space plus space left parenthesis straight n space straight h space minus space straight h right parenthesis space left parenthesis straight n space straight h space right parenthesis plus 1 half left parenthesis straight n space straight h space minus space straight h right parenthesis space left parenthesis space straight n space straight h right parenthesis space left parenthesis 2 space straight n space straight h space minus space straight h right parenthesis close square brackets$
   $equals space Lt with straight h rightwards arrow 0 below space open square brackets 5 cross times 3 plus 3 space left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis space plus space left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis space plus space 1 half left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis space left parenthesis 6 minus straight h right parenthesis close square brackets$
   $space space space space open square brackets because space space straight n space straight h space equals space straight b space minus space straight a space equals space 4 space minus space 1 space equals space 3 close square brackets$
   $equals space 15 plus 9 space left parenthesis 3 minus 0 right parenthesis plus 3 space left parenthesis 3 minus 0 right parenthesis space left parenthesis 2 right parenthesis space plus space 3 over 2 left parenthesis 3 minus 0 right parenthesis space left parenthesis 6 minus 0 right parenthesis space equals space 15 plus 27 plus 9 plus 27 space equals space 78$

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Integrals

Evaluate as limit of sums $integral subscript 1 superscript 4 left parenthesis 3 straight x squared plus 2 straight x right parenthesis space dx.$

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