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

Solution

Comparing $integral subscript 0 superscript 3 left parenthesis straight x squared plus 2 straight x right parenthesis space dx$ with $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx$ ,
$straight f left parenthesis straight x right parenthesis space equals space straight x squared plus 2 straight x comma space space space space straight a space equals space 0 comma space space straight b space equals space 3$
$straight f left parenthesis straight a right parenthesis space equals space 0 straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis straight h right parenthesis space equals space straight h squared plus 2 straight h straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 2 straight h right parenthesis space equals space 2 squared straight h squared plus 2. space 2 straight h$
........ ............. .......... ........... ........

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

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Integrals

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

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