Evaluate  as the limit of sums. from Mathematics Integrals

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 Multiple Choice QuestionsShort Answer Type

1.

Evaluate integral from 1 to 2 of straight x space dx as the limit of a sum.

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2. Evaluate the following definite integrals as limit of a sum.
integral from straight a to straight b of straight x space dx

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3. Evaluate the following definite integral as limit of a sum.
integral subscript 1 superscript 2 xdx



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4. Evaluate the following definite integral as limit of a sum.
integral subscript 0 superscript 5 left parenthesis straight x minus 1 right parenthesis dx


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5. Evaluate the following integral as the limit of a sum:
integral subscript 1 superscript 3 left parenthesis 2 straight x plus 3 right parenthesis dx
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6. Evaluate
integral subscript straight a superscript straight b straight x squared space dx  as the limit of sum.
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 Multiple Choice QuestionsLong Answer Type

7.

Evaluate  integral subscript 1 superscript 3 straight x squared space dx as the limit of a sum.

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8.

Evaluate integral subscript 1 superscript 4 left parenthesis straight x squared minus straight x right parenthesis space dx  as the limit of a sum.

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9.

Evaluate integral subscript 1 superscript 2 space left parenthesis 3 straight x squared plus 5 straight x right parenthesis space dx as limit of sums.

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10.

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


Comparing integral subscript 1 superscript 3 space 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 space we space get comma
                     straight f left parenthesis straight x right parenthesis space equals space 3 straight x squared plus 2 straight x comma space space straight a space equals space 1 comma 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 3 plus 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 space plus space straight f left parenthesis straight a plus straight h right parenthesis space plus space 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 right parenthesis space straight h close square brackets
        equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket space open curly brackets 3 plus 2 right curly bracket plus left curly bracket 3 space left parenthesis 1 plus straight h right parenthesis squared plus 2 space left parenthesis 1 plus straight h right parenthesis right curly bracket plus left curly bracket 3 space left parenthesis 1 plus 2 straight h right parenthesis squared plus 2 space left parenthesis 1 plus 2 straight h right parenthesis close curly brackets
space
                                                            plus.... plus space left curly bracket 3 space 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 space left square bracket 3 straight n plus 2 straight n plus 6 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 space left curly bracket 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right curly bracket
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space plus space 3 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 right square bracket

        equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets 5 space straight n plus 6 space straight h space fraction numerator left parenthesis straight n minus 1 right parenthesis space 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 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 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 space straight n space straight h right parenthesis plus space 1 half left parenthesis straight n space straight h minus 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 close square brackets

         equals space Lt with straight h rightwards arrow 0 below open square brackets 5 cross times 2 plus 3 space left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space plus left parenthesis 2 minus straight h right parenthesis space left parenthesis 2 right parenthesis space plus space 1 half 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

                                                                        open square brackets because space straight n space straight h space equals space straight b space minus straight a space equals space 3 space minus space 1 space equals space 2 close square brackets
         equals space 10 plus 6 space left parenthesis 2 minus 0 right parenthesis space plus space 2 left parenthesis 2 minus 0 right parenthesis space plus left parenthesis 2 minus 0 right parenthesis space left parenthesis 4 minus 0 right parenthesis
equals space 10 plus 12 plus 4 plus 8 space equals space 34
                                 

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