Evaluate the following integrals as the limit of a sum from Mat

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 Multiple Choice QuestionsLong Answer Type

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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


Comparing integral subscript 0 superscript 3 left parenthesis straight x squared plus 2 straight x right parenthesis space dx with integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx
       straight f left parenthesis straight x right parenthesis space equals space straight x squared plus 2 straight x comma space space space space straight a space equals space 0 comma space space straight b space equals space 3
     straight f left parenthesis straight a right parenthesis space equals space 0
straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis straight h right parenthesis space equals space straight h squared plus 2 straight h
straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 2 straight h right parenthesis space equals space 2 squared straight h squared plus 2. space 2 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 x rightwards arrow 0 below space straight h space 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 a 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 close square brackets
therefore     integral subscript 0 superscript 3 left parenthesis straight x squared plus 2 straight x right parenthesis dx space equals space Lt with straight h rightwards arrow 0 below straight h left square bracket 0 plus left parenthesis straight h squared plus 2 straight h right parenthesis plus left parenthesis 2 squared. straight h squared space plus 2. space 2 straight h right parenthesis
                                                               plus left parenthesis 3 squared. straight h squared plus 3. space 2 straight h right parenthesis plus open curly brackets left parenthesis straight n minus 1 right parenthesis squared. space straight h squared plus left parenthesis straight n minus 1 right parenthesis. space 2 straight h close curly brackets]

           equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets open curly brackets 1 squared plus 2 squared plus 3 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared close curly brackets space straight h squared plus space open curly brackets 1 plus 2 plus 3 plus.. plus left parenthesis straight n minus 1 right parenthesis close curly brackets. space 2 straight h close square brackets
           equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis space left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction. space straight h squared plus space fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction. space 2 straight h close square brackets

          equals Lt with straight h rightwards arrow 0 below open square brackets fraction numerator left parenthesis straight n minus straight h right parenthesis space left parenthesis nh right parenthesis space left parenthesis 2 nh minus straight h right parenthesis over denominator 6 end fraction plus left parenthesis nh minus straight h right parenthesis space left parenthesis nh right parenthesis close square brackets
        equals space Lt with straight h rightwards arrow 0 below open square brackets fraction numerator left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis space left parenthesis 6 minus straight h right parenthesis over denominator 6 end fraction plus left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis close square brackets                                     open square brackets because space space nh space equals space straight b minus straight a space equals 3 minus 0 space equals space 3 close square brackets
          equals space fraction numerator left parenthesis 3 minus 0 right parenthesis space left parenthesis 3 right parenthesis space left parenthesis 6 minus 0 right parenthesis over denominator 6 end fraction plus left parenthesis 3 minus 0 right parenthesis space left parenthesis 3 right parenthesis space equals space 9 plus 9 space equals space 18

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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.

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