Evaluate the following definite integrals as limit of a sum.
$integral from straight a to straight b of straight x space dx$

Solution

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

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Integrals

Evaluate the following definite integrals as limit of a sum.
$integral from straight a to straight b of straight x space dx$

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