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

Solution

Comparing $integral subscript 0 superscript 2 left parenthesis straight x squared plus 3 right parenthesis space dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma$ we get,
   $straight f left parenthesis straight x right parenthesis space equals space straight x squared plus 3 comma space space straight a space equals space 0 comma space space straight b space equals space 2 straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 0 right parenthesis space equals space 0 plus 3 space equals space 3$
$straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 0 plus straight h right parenthesis space equals space straight f left parenthesis straight h right parenthesis space equals space straight h squared plus 3 straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 0 plus 2 straight h right parenthesis space equals space straight f left parenthesis 2 straight h right parenthesis space equals space 2 squared. space straight h squared plus 3 ....... space space space space space space space space space space space space...... space space space space space space space space space space space...... space space space space space space space space space.....$
   $straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space straight f left parenthesis stack straight n minus 1 with bar on top space straight h right parenthesis space equals space left parenthesis straight n minus 1 right parenthesis squared space straight h squared space plus space 3$
Now    $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis space plus space straight f left parenthesis straight a plus straight h right parenthesis space plus space straight f left parenthesis straight a plus 2 straight h right parenthesis space plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis$
$therefore$    $integral subscript 0 superscript 2 left parenthesis straight x squared plus 3 right parenthesis dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket 3 plus open curly brackets straight h squared plus 3 close curly brackets space plus open curly brackets 2 squared. space straight h squared space plus space 3 close curly brackets space plus space................$
   $plus space open curly brackets left parenthesis straight n minus 1 right parenthesis squared. space straight h squared space plus space 3 close curly brackets right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets 3 space straight n space plus space straight h squared space open curly brackets 1 squared plus 2 squared plus 3 squared plus..... plus left parenthesis straight n minus 1 right parenthesis squared close curly brackets close square brackets equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets 3 space straight n space plus space straight h squared. space fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below open square brackets 3 left parenthesis straight n space straight h right parenthesis space plus space fraction numerator left parenthesis straight n space straight h right parenthesis space left parenthesis nh space minus straight h right parenthesis space left parenthesis space 2 space straight n space straight h space minus space straight h right parenthesis over denominator 6 end fraction close square brackets$
$equals space Lt with straight h space rightwards arrow 0 below space open square brackets 3 space left parenthesis 2 space right parenthesis plus space fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus straight h right parenthesis space left parenthesis 4 minus straight h right parenthesis over denominator 6 end fraction close square brackets space space space space left square bracket because space space straight n space straight h space equals space straight b minus straight a space equals space 2 minus 0 space equals space 2 right square bracket$
$equals space 3 left parenthesis 2 right parenthesis plus fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis 2 minus 0 right parenthesis space left parenthesis 4 minus 0 right parenthesis over denominator 6 end fraction space equals space 6 plus 8 over 3 space equals space 26 over 3$

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Integrals

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

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