Evaluate $integral subscript straight a superscript straight b space sinx space dx$ as the limit of a sum.

Solution

Comparing $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space with space integral subscript straight a superscript straight b sinx space dx space comma space we space get comma space space straight f left parenthesis straight x right parenthesis space equals space sin space straight x space$
Now, $integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis plus... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$therefore$ $integral subscript straight a superscript straight b sinx space dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket sin space straight a space plus sin space left parenthesis straight a plus straight h right parenthesis plus sin left parenthesis straight a plus 2 straight h right parenthesis plus... sin left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket$
$equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets fraction numerator sin space open parentheses straight a plus begin display style fraction numerator nh minus straight h over denominator 2 end fraction end style close parentheses space sin begin display style nh over 2 end style over denominator sin begin display style straight h over 2 end style end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below space open parentheses fraction numerator begin display style straight h over 2 end style over denominator sin space begin display style straight h over 2 end style end fraction close parentheses space left square bracket 2 space sin space open parentheses straight a plus fraction numerator nh minus straight h over denominator 2 end fraction close parentheses. sin straight h over 2 right square bracket$
$equals Lt with straight h rightwards arrow 0 below open parentheses fraction numerator begin display style straight h over 2 end style over denominator sin begin display style straight h over 2 end style end fraction close parentheses space open square brackets 2 space sin space open parentheses straight a plus fraction numerator straight b minus straight a minus straight h over denominator 2 end fraction close parentheses space sin space fraction numerator straight b minus straight a over denominator 2 end fraction close square brackets$ $open square brackets because space straight n space straight h space equals space straight b minus straight a close square brackets$
$equals space left parenthesis 1 right parenthesis space open square brackets 2 space sin space open parentheses straight a plus fraction numerator straight b minus straight a minus 0 over denominator 2 end fraction close parentheses space sin space fraction numerator straight b minus straight a over denominator 2 end fraction close square brackets space equals 2 space sin fraction numerator straight b plus straight a over denominator 2 end fraction sin fraction numerator straight b minus straight a over denominator 2 end fraction equals cosa minus cosb$

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Integrals

Evaluate $integral subscript straight a superscript straight b space sinx space dx$ as the limit of a sum.

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