Evaluate  as the limit of a sum. from Mathematics Integrals

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 Multiple Choice QuestionsLong Answer Type

21.

Evaluate the following integrals as the limit of a sum
integral subscript 0 superscript 3 left parenthesis straight x squared plus 2 straight x right parenthesis space dx

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22.

Evaluate the following integrals as the limit of a sum
integral subscript 0 superscript 3 left parenthesis 2 straight x squared plus 3 right parenthesis space dx

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23.

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

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24. Evaluate
integral subscript straight a superscript straight b space straight e to the power of straight x space dx as limit of sum. 
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25.

Evaluate integral subscript negative 1 end subscript superscript 1 space straight e to the power of straight x space dx as the limit of a sum.

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26.

Evaluate integral subscript 0 superscript 2 straight e to the power of straight x space dx as the limit of a sum.

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27.

Evaluate integral subscript 0 superscript 1 straight e to the power of 2 minus 3 straight x end exponent space dx as a limit of a sum.

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 Multiple Choice QuestionsShort Answer Type

28.

Evaluate integral subscript 0 superscript 3 straight e to the power of straight x space dx as the  limit of sums.

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29.

Evaluate integral subscript 0 superscript 4 left parenthesis straight x plus straight e to the power of 2 straight x end exponent right parenthesis space dx as the limit of a sum.

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30.

Evaluate integral subscript straight a superscript straight b space sinx 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 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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