Evaluate the following definite integrals as limit of a sum. fr
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Evaluate integral subscript 0 superscript 4 left parenthesis straight x plus straight e to the power of 2 straight x end exponent right parenthesis space dx as the limit of a sum.


Comparing integral subscript 0 superscript 4 left parenthesis straight x plus straight e to the power of 2 straight x end exponent right parenthesis dx space space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis comma space we space get
             straight f left parenthesis straight x right parenthesis space equals space straight x plus straight e to the power of 2 straight x end exponent comma space space straight a space equals space 0 comma space space straight b space equals space 4
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 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 space straight h right parenthesis right square bracket
therefore     integral subscript 0 superscript 4 left parenthesis straight x plus straight e to the power of 2 straight x end exponent right parenthesis space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket straight f left parenthesis 0 right parenthesis plus straight f left parenthesis straight h right parenthesis plus straight f left parenthesis 2 straight h right parenthesis plus... plus straight f left parenthesis stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket

        equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket left parenthesis 0 plus straight e to the power of 0 right parenthesis space plus left parenthesis straight h plus straight e to the power of 2 straight h end exponent right parenthesis space plus space left parenthesis 2 straight h plus straight e to the power of 4 straight h end exponent right parenthesis plus... plus left curly bracket left parenthesis straight n minus 1 right parenthesis space straight h space plus straight e to the power of 2 left parenthesis straight n minus 1 right parenthesis straight h end exponent right curly bracket right square bracket
         equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket straight h left parenthesis 1 plus 2 plus 3 plus... plus stack straight n minus 1 with bar on top right parenthesis plus left parenthesis 1 plus straight e to the power of 2 straight h end exponent plus straight e to the power of 4 straight h end exponent plus... plus straight e to the power of 2 left parenthesis straight n minus 1 right parenthesis space straight h end exponent right parenthesis right square bracket
         equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets 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 fraction numerator 1 left parenthesis straight e to the power of 2 nh end exponent minus 1 right parenthesis over denominator straight e to the power of 2 straight h end exponent minus 1 end fraction close square brackets space space space space space space space space space space space space space open square brackets because straight S subscript straight n space equals space fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction comma space straight r greater than 1 close square brackets
equals space Lt with straight h rightwards arrow 0 below space straight h open square brackets fraction numerator left parenthesis straight n space straight h minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis over denominator 2 end fraction plus fraction numerator straight e to the power of 2 nh end exponent minus 1 over denominator 2 cross times begin display style fraction numerator straight e to the power of 2 straight h end exponent minus 1 over denominator 2 straight h end fraction end style end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below space open square brackets fraction numerator left parenthesis 4 minus straight h right parenthesis space left parenthesis 4 right parenthesis over denominator 2 end fraction plus fraction numerator straight e to the power of 8 minus 1 over denominator 2 cross times begin display style fraction numerator straight e to the power of 2 straight h end exponent minus 1 over denominator 2 straight h end fraction end style end fraction close square brackets

                                                                                           open square brackets because space straight n apostrophe straight h space equals space straight b minus straight a space equals space 4 minus 0 space equals space 4 close square brackets
           = fraction numerator left parenthesis 4 minus 0 right parenthesis space left parenthesis 4 right parenthesis over denominator 2 end fraction plus fraction numerator straight e to the power of 8 minus 1 over denominator 2 cross times loge end fraction space equals space 8 plus fraction numerator straight e to the power of 8 minus 1 over denominator 2 end fraction space equals space fraction numerator straight e to the power of 8 plus 15 over denominator 2 end fraction

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Evaluate the following definite integrals as limit of sums.
integral subscript 2 superscript 3 straight x squared dx


Let             I = integral subscript 2 superscript 3 straight x squared dx
Since    integral straight x squared dx space equals space straight x cubed over 3 space equals space straight F left parenthesis straight x right parenthesis comma space say.
therefore    by second fundamental theorem,
                                     I = F(3) - F(2) = fraction numerator left parenthesis 3 right parenthesis cubed over denominator 3 end fraction minus fraction numerator left parenthesis 2 right parenthesis cubed over denominator 3 end fraction space equals space 9 minus 8 over 3 space equals space 19 over 3

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


Comparing integral subscript straight a superscript straight b xdx space 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
therefore            straight f left parenthesis straight a right parenthesis space equals space straight a plus straight h
    straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight a plus 2 straight h
......................................
straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space straight a plus stack straight n minus 1 with bar on top straight h
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
                           equals space Lt with straight h rightwards arrow 0 below straight h left square bracket straight a plus left parenthesis straight a plus straight h right parenthesis plus left parenthesis straight a plus 2 straight h right parenthesis plus.... plus left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis right square bracket
equals space Lt with straight h rightwards arrow 0 below straight h left square bracket straight n space straight a space plus space straight h space open curly brackets 1 plus 2 plus 3 plus....... plus left parenthesis straight n minus 1 right parenthesis close curly brackets right square bracket
equals space Lt with straight h rightwards arrow 0 below straight h open square brackets straight n space straight a 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 close square brackets space equals space Lt with straight h rightwards arrow 0 below open square brackets straight a left parenthesis straight n. straight h right parenthesis space plus space fraction numerator left parenthesis straight n space straight h space minus straight h right parenthesis space left parenthesis straight n space straight h right parenthesis over denominator 2 end fraction close square brackets
equals space Lt with straight h rightwards arrow 0 below open square brackets straight a left parenthesis straight b minus straight a right parenthesis plus fraction numerator left parenthesis straight b minus straight a minus straight h right parenthesis space left parenthesis straight b minus straight a right parenthesis over denominator 2 end fraction close square brackets
equals space straight a left parenthesis straight b minus straight a right parenthesis plus fraction numerator left parenthesis straight b minus straight a minus 0 right parenthesis space left parenthesis straight b minus straight a right parenthesis over denominator 2 end fraction
equals space straight a left parenthesis straight b minus straight a right parenthesis plus fraction numerator left parenthesis straight b minus straight a right parenthesis squared over denominator 2 end fraction space equals space left parenthesis straight b minus straight a right parenthesis space open square brackets straight a plus fraction numerator straight b minus straight a over denominator 2 end fraction close square brackets
equals space left parenthesis straight b minus straight a right parenthesis open square brackets fraction numerator 2 straight a plus straight b minus straight a over denominator 2 end fraction close square brackets space equals space left parenthesis straight b minus straight a right parenthesis open parentheses fraction numerator straight b plus straight a over denominator 2 end fraction close parentheses space equals space 1 half left parenthesis straight b squared minus straight a squared right parenthesis.

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Evaluate integral subscript negative 1 end subscript superscript 1 space straight e to the power of straight x space dx as the limit of a sum.


Comparing  integral subscript negative 1 end subscript superscript 1 space straight e to the power of straight x 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 comma

               straight f left parenthesis straight x right parenthesis space equals space straight e to the power of straight x comma space space space straight a space equals space minus 1 comma space space straight b space equals space 1

therefore space space straight f left parenthesis straight a right parenthesis space equals space straight e to the power of straight a comma space space straight f left parenthesis straight a plus straight h right parenthesis space equals space straight e to the power of straight a plus straight h end exponent comma space space straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight e to the power of straight a plus 2 straight h end exponent comma space space..... comma

       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 e to the power of straight a plus left parenthesis straight n minus 1 right parenthesis straight h end exponent

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 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 space straight h right parenthesis right square bracket
therefore       integral subscript negative 1 end subscript superscript 1 space straight e to the power of straight x space dx space equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets straight e to the power of straight a plus straight e to the power of straight a plus straight h end exponent plus straight e to the power of straight a plus 2 straight h end exponent plus.... plus straight e to the power of straight a plus left parenthesis straight n minus 1 right parenthesis space straight h end exponent close square brackets
                             equals space Lt with straight h rightwards arrow 0 below straight h open square brackets fraction numerator straight e to the power of straight a left parenthesis straight e to the power of straight n space straight h end exponent minus 1 right parenthesis over denominator straight e to the power of straight h minus 1 end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below straight h open square brackets fraction numerator straight e to the power of straight a left parenthesis straight e squared minus 1 right parenthesis over denominator straight e to the power of straight h minus 1 end fraction close square brackets

                                                                             left square bracket because space space straight n space straight h space equals space straight b space minus space straight a space equals space 1 plus 1 space equals space 2 right square bracket

                            equals straight e to the power of straight a left parenthesis straight e squared minus 1 right parenthesis space Lt with straight h rightwards arrow 0 below fraction numerator straight h over denominator straight e to the power of straight h minus 1 end fraction space equals straight e to the power of straight a left parenthesis straight e squared minus 1 right parenthesis space fraction numerator 1 over denominator begin display style Lt with straight h rightwards arrow 0 below fraction numerator straight e to the power of straight h minus 1 over denominator straight h end fraction end style end fraction

                                equals space straight e to the power of straight a left parenthesis straight e squared minus 1 right parenthesis. space 1 over 1 space equals space straight e to the power of straight a left parenthesis straight e squared minus 1 right parenthesis space equals space straight e to the power of negative 1 end exponent left parenthesis straight e squared minus 1 right parenthesis space space space space space left square bracket because space straight a space equals space minus 1 right square bracket
equals space straight e minus straight e to the power of negative 1 end exponent space equals space straight e minus 1 over straight e

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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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