Evaluate the following definite integral as limit of a sum.
$integral subscript 1 superscript 2 xdx$

Solution

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

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Integrals

Evaluate the following definite integral as limit of a sum.
$integral subscript 1 superscript 2 xdx$

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