Evaluate  as the limit of a sum. from Mathematics Integrals

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 Multiple Choice QuestionsLong Answer Type

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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.


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

                        equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets 2 space straight n space plus space straight h space left parenthesis 1 plus 2 plus..... plus stack straight n minus 1 with bar on top right parenthesis space plus space straight h squared space open curly brackets 1 squared plus 2 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared close curly brackets close square brackets
                        equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets 2 space straight n space space plus straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n right parenthesis over denominator 2 end fraction plus straight h squared fraction numerator left parenthesis straight n minus 1 right parenthesis left parenthesis straight n right parenthesis left parenthesis 2 straight n minus 1 right parenthesis over denominator 6 end fraction close square brackets
equals space Lt with straight h rightwards arrow 0 below space open square brackets 2 space straight n space straight h space plus space fraction numerator left parenthesis space straight n space straight h right parenthesis space left parenthesis straight n space straight h space minus straight h right parenthesis over denominator 2 end fraction plus fraction numerator left parenthesis space straight n space straight h right parenthesis space left parenthesis straight n space straight h space minus straight h right parenthesis space left parenthesis 2 space straight n space straight h space minus straight h right parenthesis over denominator 6 end fraction close square brackets
equals space Lt with straight h rightwards arrow 0 below space open square brackets 4 plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus straight h right parenthesis over denominator 2 end fraction plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus straight h right parenthesis space left parenthesis 4 minus straight h right parenthesis over denominator 6 end fraction close square brackets

                                                                             open square brackets because space space straight n space straight h space equals space straight b minus straight a space equals space 2 minus 0 space equals space 2 close square brackets

                                    equals space 4 plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus 0 right parenthesis over denominator 2 end fraction plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus 0 right parenthesis space left parenthesis 4 minus 0 right parenthesis over denominator 6 end fraction
equals space 4 plus 2 plus 8 over 3 space equals fraction numerator 12 plus 6 plus 8 over denominator 3 end fraction space equals 26 over 3

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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

14.

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

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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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