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.

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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


Comparing integral subscript 1 superscript 3 left parenthesis 2 straight x squared plus 3 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 2 straight x squared plus 3 straight x comma space space straight a space equals space 1 comma space space space straight b space equals space 3
straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 2 space plus 3
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 2 left parenthesis 1 plus straight h right parenthesis squared plus 3 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 2 left parenthesis 1 plus 2 straight h right parenthesis squared plus 3 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 space 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 2 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared plus 3 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 space straight h right parenthesis close square brackets
    equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket open curly brackets 2 plus 3 close curly brackets space plus space left curly bracket 2 space left parenthesis 1 plus straight h right parenthesis squared space plus space 3 space left parenthesis 1 plus straight h right parenthesis right curly bracket space plus space left curly bracket 2 left parenthesis 1 plus 2 straight h right parenthesis squared plus 3 left parenthesis 1 plus 2 straight h right parenthesis space right curly bracket      
                                                                      plus.... plus left curly bracket 2 left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared space plus space 3 space 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 2 space straight n space plus space 3 space straight n space plus space 4 space straight h space left square bracket 1 plus 2 plus........ plus stack straight n minus 1 with bar on top right curly bracket space plus space 3 space straight h space left curly bracket 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right curly bracket
                                                                                    plus 2 space straight h squared space 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

      space equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets 5 space straight n plus space 4 space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space straight n over denominator 2 end fraction plus 3 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 2 space straight h squared space 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
equals space Lt with straight h rightwards arrow 0 below space open square brackets 5 space straight n space straight h space plus space 2 space left parenthesis space straight n space straight h space minus straight h right parenthesis space left parenthesis space straight n space straight h right parenthesis space plus space 3 over 2 left parenthesis straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis space plus space 1 third left parenthesis space straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis thin space left parenthesis 2 space straight n space straight h space minus straight h right parenthesis close square brackets
      equals space Lt with straight h rightwards arrow 0 below space open square brackets 5 cross times 2 plus 2 left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space plus space 3 over 2 left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space plus space 1 third left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space left parenthesis 4 minus straight h right parenthesis close square brackets
                                                                                                      left square bracket because space straight n space straight h space equals space space straight b space minus space straight a space equals space 3 space minus space 1 space equals space 2 space right square bracket
       equals 10 plus 4 space left parenthesis 2 minus 0 right parenthesis space plus space 3 over 2 left parenthesis 2 minus 0 right parenthesis space left parenthesis 2 right parenthesis space plus 1 third left parenthesis 2 minus 0 right parenthesis space left parenthesis 2 right parenthesis space left parenthesis 4 minus 0 right parenthesis
equals space 10 plus 8 plus 6 plus 16 over 3 space equals space 88 over 3
       

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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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