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31.
Evaluate $integral subscript straight a superscript straight b cosx space dx$ as the limit of a sum.

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

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Long Answer Type

32.

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

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Short Answer Type

33.

Evaluate the following definite integrals as limit of sums.
$integral subscript 2 superscript 3 straight x squared dx$

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34.

Evaluate the following definite integral
$integral subscript negative 1 end subscript superscript 1 left parenthesis straight x plus 1 right parenthesis space dx$

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35.

Evaluate the following definite integral
$integral subscript 2 superscript 3 1 over straight x dx$

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36.

Evaluate the following definite integral
$integral subscript 0 superscript 8 straight x to the power of 5 over 3 end exponent dx$

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37.

Evaluate the following definite integral
$integral subscript 0 superscript 4 straight x to the power of 1 half end exponent dx$

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38.

Evaluate the following definite integral
$integral subscript 0 superscript 4 open parentheses straight x plus straight x to the power of 3 over 2 end exponent close parentheses dx$