Evaluate the following definite integral as limit of a sum. fro

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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



Comparing integral subscript 0 superscript 5 left parenthesis straight x minus 1 right parenthesis dx space with space integral subscript straight a superscript straight b space straight f left parenthesis straight x right parenthesis space dx comma space space space we space get comma
   f(x) = x - 1,  a = 0,  b = 5
straight f left parenthesis straight a plus 2 space straight h right parenthesis space equals space straight f left parenthesis 2 space straight h right parenthesis space equals space 2 space straight h space minus space 1 comma space space..... comma space space 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 stack straight n minus 1 with bar on top straight h right parenthesis space equals space left parenthesis straight n minus 1 right parenthesis space straight h space minus 1

Now integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space equals space Lt with straight h rightwards arrow 0 below space straight h 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 stack straight n minus 1 with bar on top straight h right parenthesis close square brackets

          equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets left parenthesis negative 1 right parenthesis plus left parenthesis straight h minus 1 right parenthesis plus left parenthesis 2 straight h minus 1 right parenthesis plus.... plus open curly brackets left parenthesis straight n minus 1 right parenthesis straight h minus 1 close curly brackets close square brackets
equals space Lt with straight h space rightwards arrow 0 below space straight h open square brackets negative straight n plus straight h open curly brackets 1 plus 2 plus 3 plus... plus left parenthesis straight n minus 1 right parenthesis close curly brackets close square brackets space equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets negative straight n plus straight h fraction numerator left parenthesis straight n minus 1 minus straight n right parenthesis over denominator 2 end fraction close square brackets
      equals space Lt with straight h rightwards arrow 0 below open square brackets negative nh plus fraction numerator left parenthesis straight n space straight h right parenthesis space left parenthesis nh space minus space straight h right parenthesis over denominator 2 end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below open square brackets negative left parenthesis 5 minus 0 right parenthesis plus left parenthesis 5 minus 0 right parenthesis fraction numerator left parenthesis 5 minus 0 minus straight h right parenthesis over denominator 2 end fraction close square brackets
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 space space space space open square brackets because space space straight n space straight h space equals space straight b space minus straight a space equals space 5 space minus space 0 close square brackets
equals space minus 5 plus fraction numerator 5 left parenthesis 5 minus 0 right parenthesis over denominator 2 end fraction equals negative 5 plus 25 over 2 equals fraction numerator negative 10 plus 25 over denominator 2 end fraction equals 15 over 2
                     

                     

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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.

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