Evaluate the following integrals as the limit of a sum
$integral subscript 0 superscript 3 left parenthesis 2 straight x squared plus 3 right parenthesis space dx$

Solution

Comparing $integral subscript 0 superscript 3 left parenthesis 2 straight x squared plus 3 right parenthesis space dx space space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space we space get comma$
   $straight f left parenthesis straight x right parenthesis space equals space 2 straight x squared plus 3 comma space space space space space straight a space equals space 0 comma space space space straight b space equals space 3 straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 0 right parenthesis space equals space 0 plus 3 space equals space 3 straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis straight h right parenthesis space equals space 2 straight h squared plus 3$
   $straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 2 straight h right parenthesis space space equals space 2. space 2 squared space straight h squared plus 3$
......... ........... ........... .......... .......... .............
$straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight f left parenthesis stack straight n minus 1 with bar on top space straight h right parenthesis space equals space 2. left parenthesis straight n minus 1 right parenthesis squared straight h squared plus 3$
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 space 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 space straight h right parenthesis close square brackets space space space space space space space space space space space space space space space space space space equals space stack Lt space straight h with straight h rightwards arrow 0 below space open square brackets 3 plus left parenthesis 2 straight h squared plus 3 right parenthesis plus left parenthesis 2.2 squared straight h squared plus 3 right parenthesis plus..... plus open curly brackets 2 left parenthesis straight n minus 1 right parenthesis squared space straight h squared plus 3 close curly brackets close square brackets space space space space space space space space space space space space space space space space space space space equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets 3 straight n plus 2 straight h squared left curly bracket 1 squared plus 2 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared right curly bracket close square brackets space space space space space space space space space space space space space space space space space space equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets 3 straight n plus 2 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 space space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below space open square brackets 3 space nh space plus space fraction numerator left parenthesis straight n minus straight h right parenthesis space left parenthesis nh right parenthesis space left parenthesis 2 nh minus straight h right parenthesis over denominator 3 end fraction close square brackets$
$equals space Lt with straight h rightwards arrow 0 below space open square brackets 3 left parenthesis 3 right parenthesis space plus space fraction numerator left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis thin space left parenthesis 6 minus straight h right parenthesis over denominator 3 end fraction close square brackets space space space space space left square bracket because space straight n space straight h space equals space straight b minus straight a space equals space 3 minus 0 space equals space 3 right square bracket equals space space space 9 plus fraction numerator left parenthesis 3 minus 0 right parenthesis space left parenthesis 3 right parenthesis thin space left parenthesis 6 minus 0 right parenthesis over denominator 3 end fraction space equals space 9 plus 18 space equals space 27$

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Integrals

Evaluate the following integrals as the limit of a sum
$integral subscript 0 superscript 3 left parenthesis 2 straight x squared plus 3 right parenthesis space dx$

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