Evaluate $integral subscript 2 superscript 3 space straight x cubed space dx$ as the limit of a sum.

Solution

Comparing $integral subscript 2 superscript 3 space straight x cubed space dx space space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space we space get$
$straight f left parenthesis straight x right parenthesis space equals space straight x cubed comma space space straight a space equals space 2 comma space space straight b space equals space 3 straight f left parenthesis straight a right parenthesis space equals space 2 cubed straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 2 plus 2 space straight h right parenthesis space equals space left parenthesis 2 plus 2 straight h right parenthesis cubed$
.......................................................
$straight f left parenthesis straight a plus stack straight n minus 1 with bar on top right parenthesis space equals space straight f left parenthesis 2 plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space left parenthesis 2 plus stack straight n minus 1 with bar on top space straight h right parenthesis cubed$
Now, $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis$
$equals stack space Lt with straight h rightwards arrow 0 below space straight h space left square bracket 2 cubed plus left parenthesis 2 plus straight h right parenthesis cubed plus left parenthesis 2 plus 2 straight h right parenthesis cubed plus.... plus left parenthesis 2 plus stack straight n minus 1 with bar on top space straight h right parenthesis cubed right square bracket$
$equals space stack Lt space straight h with straight h rightwards arrow 0 below space left square bracket 2 cubed space cross times space straight n space plus space 12 space straight h space left curly bracket 1 plus 2 plus 3.. plus.... left parenthesis straight n minus 1 right parenthesis right curly bracket space plus space 6 straight h squared space left curly bracket 1 squared plus 2 squared plus 3 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared right square bracket$
$plus straight h cubed space left curly bracket 1 cubed plus 2 cubed plus 3 cubed plus... plus left parenthesis straight n minus 1 right parenthesis cubed right curly bracket right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets 8 space straight n plus 12 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction plus 6 space straight h squared fraction numerator left parenthesis straight n minus 1 right parenthesis left parenthesis straight n right parenthesis left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction plus straight h cubed fraction numerator left parenthesis straight n minus 1 right parenthesis squared space straight n squared over denominator 4 end fraction close square brackets equals space Lt with straight h rightwards arrow 0 below open square brackets 8 space nh space plus space 6 left parenthesis straight n space straight h space minus space straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space plus space left parenthesis straight n space straight h minus straight h right parenthesis space left parenthesis nh right parenthesis space left parenthesis 2 space straight n space straight h space minus straight h right parenthesis space plus space space fraction numerator left parenthesis nh minus straight h right parenthesis squared space left parenthesis straight n space straight h right parenthesis squared over denominator 4 end fraction close square brackets$
$equals space Lt with straight h rightwards arrow 0 below space open square brackets 8 cross times 1 plus 6 left parenthesis 1 minus straight h right parenthesis space left parenthesis 1 right parenthesis space plus space left parenthesis 1 minus straight h right parenthesis space left parenthesis 1 right parenthesis space left parenthesis 2 minus straight h right parenthesis plus fraction numerator left parenthesis 1 minus straight h right parenthesis squared left parenthesis 1 right parenthesis squared over denominator 4 end fraction close square brackets$ $left square bracket because space straight n space straight h space equals space straight b minus straight a space equals space 3 minus 2 space equals space 1 right square bracket$
$equals 8 plus 6 space left parenthesis 1 minus 0 right parenthesis space left parenthesis 1 right parenthesis space plus left parenthesis 1 minus 0 right parenthesis space left parenthesis 1 right parenthesis space left parenthesis 2 minus 0 right parenthesis space plus space fraction numerator left parenthesis 1 minus 0 right parenthesis squared space left parenthesis 1 right parenthesis squared over denominator 4 end fraction space equals space 8 plus 6 plus 2 plus 1 fourth space equals space 65 over 4$

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Integrals

Evaluate $integral subscript 2 superscript 3 space straight x cubed space dx$ as the limit of a sum.

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