Evaluate  as the limit of a sum. from Mathematics Integrals

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 Multiple Choice QuestionsLong Answer Type

11.

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

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12.

Evaluate integral subscript 0 superscript 2 left parenthesis straight x squared plus straight x plus 1 right parenthesis space dx as the limit of a sum.

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13.

Evaluate  integral subscript 1 superscript 3 left parenthesis straight x squared plus straight x right parenthesis space dx as the limit of sums. 

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 Multiple Choice QuestionsShort Answer Type

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14.

Evaluate integral subscript 1 superscript 2 left parenthesis straight x squared plus straight x right parenthesis dx as the limit of a sum.


Comparing integral subscript 1 superscript 2 left parenthesis straight x squared plus straight x right parenthesis space dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma space we get,
          straight f left parenthesis straight x right parenthesis space equals space straight x squared plus straight x comma space space straight a space equals space 1 comma space space straight b space equals space 2
therefore                   straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals 1 plus 1
                   straight f left parenthesis straight a plus straight h right parenthesis space equals space straight f left parenthesis 1 plus straight h right parenthesis space equals left parenthesis 1 plus straight h right parenthesis squared plus left parenthesis 1 plus straight h right parenthesis
straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight f left parenthesis 1 plus 2 straight h right parenthesis space equals space left parenthesis 1 plus 2 straight h right parenthesis squared plus left parenthesis 1 plus 2 straight h right parenthesis
............................................................................
        straight f left parenthesis straight a plus stack straight n minus 1 with bar on top right parenthesis space equals space straight f left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis space equals space left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis
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 space straight h right parenthesis right square bracket

therefore   integral subscript 1 superscript 2 left parenthesis straight x squared plus straight x right parenthesis dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket left curly bracket 1 plus 1 right curly bracket plus left curly bracket left parenthesis 1 plus straight h right parenthesis squared plus left parenthesis 1 plus straight h right parenthesis right curly bracket plus left curly bracket left parenthesis 1 plus 2 straight h right parenthesis squared plus left parenthesis 1 plus 2 straight h right parenthesis right curly bracket plus.....  

                                                                               ... plus left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus left parenthesis 1 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 left square bracket straight n plus 2 straight h space left curly bracket 1 plus 2 plus 3 plus....... plus stack straight n minus 1 with bar on top right curly bracket space plus space straight h squared left curly bracket 1 squared plus 2 squared plus 3 squared plus.....    
                                                                plus left parenthesis straight n minus 1 right parenthesis squared right curly bracket plus straight n plus straight h space left curly bracket 1 plus 2 plus 3 plus.... plus stack straight n minus 1 with bar on top right curly bracket right square bracket

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

                                 equals 1 plus left parenthesis 1 minus 0 right parenthesis space left parenthesis 1 right parenthesis space plus space fraction numerator left parenthesis 1 minus 0 right parenthesis left parenthesis 1 right parenthesis space left parenthesis 2 minus 0 right parenthesis over denominator 6 end fraction plus 1 plus fraction numerator left parenthesis 1 minus 0 right parenthesis space left parenthesis 1 right parenthesis over denominator 2 end fraction
equals space 1 plus 1 plus 1 third plus 1 plus 1 half space equals 23 over 3

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15.

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

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 Multiple Choice QuestionsLong Answer Type

16.

Evaluate as limit of sums integral subscript 1 superscript 4 left parenthesis 5 straight x squared plus 3 straight x right parenthesis space dx.

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17.

Evaluate as limit of sums integral subscript 1 superscript 4 left parenthesis 3 straight x squared plus 2 straight x right parenthesis space dx.

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18.

Evaluate as limit of sums: integral subscript 1 superscript 3 left parenthesis 2 straight x squared plus 3 straight x right parenthesis space dx. space

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19. Evaluate the following integral as limit of a sum
integral subscript 0 superscript 2 left parenthesis straight x squared plus 3 right parenthesis space dx.
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20.

Evaluate the following integrals as the limit of a sum
integral subscript 0 superscript 2 left parenthesis straight x squared plus 1 right parenthesis space dx

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