Evaluate as limit of sums  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

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.


Comparing  integral subscript 1 superscript 4 left parenthesis 3 straight x squared plus 2 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 space get
                   straight f left parenthesis straight x right parenthesis space equals space 3 straight x squared plus 2 straight x comma space space space straight a space equals space 1 comma space space straight b space equals space 4
straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 3 space plus space 2
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 space 3 left parenthesis 1 plus straight h right parenthesis squared plus 2 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 3 left parenthesis 1 plus 2 straight h right parenthesis squared plus 2 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 straight h 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 3 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus 2 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 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 f left parenthesis straight a plus 2 straight h right parenthesis plus.... plus straight f left parenthesis straight a plus top enclose straight n minus 1 end enclose right parenthesis straight h close square brackets
               equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket space left curly bracket 3 plus 2 right curly bracket plus left curly bracket 3 left parenthesis 1 plus straight h right parenthesis squared plus 2 left parenthesis 1 plus straight h right parenthesis right curly bracket space plus space left curly bracket 3 space left parenthesis 1 plus 2 straight h right parenthesis squared plus 2 left parenthesis 1 plus 2 straight h right parenthesis right curly bracket
                                                                    plus..... plus left curly bracket 3 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared space plus space 2 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis right curly bracket right square bracket
                equals space Lt with straight h rightwards arrow 0 below left square bracket 3 space straight n space plus space 2 space straight n space plus space 6 space straight h space left curly bracket 1 plus 2 plus..... plus stack straight n minus 1 with bar on top right curly bracket space plus space 2 straight h left curly bracket 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right curly bracket
                                                                          plus 3 straight h squared left curly bracket 1 squared plus 2 squared plus..... left parenthesis stack straight n minus 1 with bar on top right parenthesis squared right curly bracket right square bracket
                 equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets 5 straight n space plus 6 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis straight n over denominator 2 end fraction plus 2 space 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 3 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 close square brackets space
equals space Lt with straight h rightwards arrow 0 below space open square brackets 5 space straight n space straight h space plus space 3 space left parenthesis straight n space straight h space minus space straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space plus space left parenthesis straight n space straight h space minus space straight h right parenthesis space left parenthesis straight n space straight h space right parenthesis plus 1 half left parenthesis straight n space straight h space minus space straight h right parenthesis space left parenthesis space straight n space straight h right parenthesis space left parenthesis 2 space straight n space straight h space minus space straight h right parenthesis close square brackets
                  equals space Lt with straight h rightwards arrow 0 below space open square brackets 5 cross times 3 plus 3 space left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis space plus space left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis space plus space 1 half left parenthesis 3 minus straight h right parenthesis space left parenthesis 3 right parenthesis space left parenthesis 6 minus straight h right parenthesis close square brackets
                                                                                      space space space space open square brackets because space space straight n space straight h space equals space straight b space minus space straight a space equals space 4 space minus space 1 space equals space 3 close square brackets
                equals space 15 plus 9 space left parenthesis 3 minus 0 right parenthesis plus 3 space left parenthesis 3 minus 0 right parenthesis space left parenthesis 2 right parenthesis space plus space 3 over 2 left parenthesis 3 minus 0 right parenthesis space left parenthesis 6 minus 0 right parenthesis
space equals space 15 plus 27 plus 9 plus 27 space equals space 78

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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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