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

Comparing integral subscript straight a superscript straight b straight x squared dx with 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 straight x squared
therefore                     straight f left parenthesis straight a right parenthesis space equals space straight a squared
               straight f left parenthesis straight a plus straight h right parenthesis space equals space left parenthesis straight a plus straight h right parenthesis squared
straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space left parenthesis straight a plus 2 straight h right parenthesis squared
              .............................
         straight f left parenthesis straight a plus stack straight n minus 1 with bar on top minus straight h right parenthesis space equals space left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis squared
Now       integral subscript straight a superscript straight b space 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 straight h right parenthesis right square bracket
where      nh space equals space straight b minus straight a
therefore          integral subscript straight a superscript straight b straight x squared dx space equals space Lt with straight h rightwards arrow 0 below straight h left square bracket straight a squared plus left parenthesis straight a plus straight h right parenthesis squared plus left parenthesis straight a plus 2 straight h right parenthesis squared plus.... plus left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis squared
         equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket left parenthesis straight a squared plus straight a squared plus straight a squared plus..... space to space straight n space term right parenthesis space plus space 2 space straight a space straight h space left curly bracket 1 plus 2 plus 3 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
                                                                          plus straight h squared left curly bracket 1 squared plus 2 squared plus 3 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared right curly bracket right square bracket
equals space Lt with straight h rightwards arrow 0 below space straight h space space open square brackets na squared plus 2 ah. fraction numerator left parenthesis straight n minus 1 right parenthesis straight n over denominator 2 end fraction plus straight h squared fraction numerator left parenthesis straight n minus 1 right parenthesis left parenthesis straight n right parenthesis 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 open square brackets straight a squared nh plus straight a left parenthesis nh minus straight h right parenthesis left parenthesis nh right parenthesis plus fraction numerator left parenthesis nh minus straight h right parenthesis left parenthesis nh right parenthesis left parenthesis 2 nh minus straight h right parenthesis over denominator 6 end fraction close square brackets
equals space Lt with straight h rightwards arrow 0 below open square brackets straight a squared left parenthesis straight b minus straight a right parenthesis plus straight a left parenthesis straight a minus straight b minus straight h right parenthesis space 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 left parenthesis 2 straight b minus 2 straight a minus straight h right parenthesis over denominator 6 end fraction close square brackets
equals space straight a squared left parenthesis straight b minus straight a right parenthesis plus straight a left parenthesis straight b minus straight a minus 0 right parenthesis left parenthesis straight b minus straight a right parenthesis plus fraction numerator left parenthesis straight b minus straight a minus 0 right parenthesis left parenthesis straight b minus straight a right parenthesis left parenthesis 2 straight b minus 2 straight a minus 0 right parenthesis over denominator 6 end fraction
equals space straight a squared left parenthesis straight b minus straight a right parenthesis plus straight a left parenthesis straight b minus straight a right parenthesis squared plus 1 third left parenthesis straight b minus straight a right parenthesis cubed space equals space left parenthesis straight b minus straight a right parenthesis open square brackets straight a squared plus straight a left parenthesis straight b minus straight a right parenthesis plus 1 third left parenthesis straight b minus straight a right parenthesis squared close square brackets
equals space left parenthesis straight b minus straight a right parenthesis open square brackets fraction numerator 3 straight a squared plus 3 straight a left parenthesis straight b minus straight a right parenthesis plus left parenthesis straight b minus straight a right parenthesis squared over denominator 3 end fraction close square brackets
equals space fraction numerator left parenthesis straight b minus straight a right parenthesis over denominator 3 end fraction left square bracket 3 space straight a squared plus 3 space straight a space straight b space minus space 3 space straight a squared space plus space straight b squared space plus space straight a squared space minus space 2 space straight a space straight b right square bracket
equals space 1 third left parenthesis straight b minus straight a right parenthesis space left parenthesis straight a squared plus straight b squared plus ab right parenthesis space equals space 1 third left parenthesis straight b cubed minus straight a cubed right parenthesis


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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

Comparing integral subscript 1 superscript 3 left parenthesis 2 straight x plus 3 right parenthesis dx space with space integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx comma we get
         f(x) = 2x +3 ,   a = 1,  b = 3
therefore    straight f left parenthesis straight a right parenthesis space equals space straight f left parenthesis 1 right parenthesis space equals space 2 plus 3 space equals space 5 comma space space 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 space plus space 3 space equals space 5 space plus space 2 straight h
         f(a+2h) = f(1+2h) = 2(1+2h)+3 = 5+4h
          ...     ...      ...        ...         ...          ...
        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 straight h right parenthesis space equals space 2 open curly brackets 1 plus left parenthesis straight n minus 1 right parenthesis space straight h close curly brackets space plus 3 space equals space 5 plus 2 left parenthesis straight n minus 1 right parenthesis space straight h

Now, integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis dx space equals space Lt with straight h rightwards arrow 0 below 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 space straight h right parenthesis close square brackets

rightwards double arrow space space space integral subscript 1 superscript 3 left parenthesis 2 straight x plus 3 right parenthesis dx space equals space Lt with straight h rightwards arrow 0 below straight h left square bracket 5 plus left parenthesis 5 plus 2 straight h right parenthesis plus left parenthesis 5 plus 4 straight h right parenthesis plus.... plus open curly brackets 5 plus 2 left parenthesis straight n minus 1 right parenthesis space straight h close curly brackets
                                  equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket 5 straight n plus 2 straight h 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 space straight h open square brackets 5 straight n plus 2 straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space straight n over denominator 2 end fraction close square brackets space equals space Lt with straight h rightwards arrow 0 below open square brackets 5 space straight n space straight h space plus left parenthesis straight n space straight h right parenthesis space left parenthesis nh space minus straight h right parenthesis close square brackets
equals space Lt with straight h rightwards arrow 0 below space left square bracket space 5 space left parenthesis 2 right parenthesis space plus space left parenthesis 2 right parenthesis space left parenthesis 2 minus straight h right parenthesis right square bracket space space space space space space space space space space space space space space space space open square brackets because space straight n space straight h space space equals space straight b space minus straight a space equals space 3 space minus space 1 space equals space 2 close square brackets
space equals space 5 space left parenthesis 2 right parenthesis space plus space 2 left parenthesis 2 minus 0 right parenthesis space equals space 10 plus 4 space equals space 14

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


Comparing integral from 1 to 2 of straight x space dx space space with space integral from straight a to straight b of 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 straight x comma space space space straight a space equals space 1 comma space space straight b space equals space 2
therefore space space space straight f left parenthesis straight a right parenthesis space equals space straight a comma space space straight f left parenthesis straight a plus straight h right parenthesis space equals space straight a plus straight h comma space space straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight a plus 2 straight h comma space...... 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 left parenthesis straight n minus 1 right parenthesis straight h

Now   integral from straight a to straight b of straight f left parenthesis straight x right parenthesis space equals space stack Lt. straight h with straight h rightwards arrow 0 below left square bracket space 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 space plus space....... 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
                                                                                              where n h = b - a
rightwards double arrow        integral from straight a to straight b of xdx space equals space Lt with straight h rightwards arrow 0 below straight h left square bracket straight a plus straight a left parenthesis straight a plus straight h right parenthesis plus left parenthesis straight a plus 2 space straight h right parenthesis space plus space... plus left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis right square bracket

                          equals stack Lt space straight h with straight h rightwards arrow 0 below left square bracket left parenthesis straight a plus straight a plus straight a plus... to space straight n space terms right parenthesis space plus space straight h 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 na plus straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n minus 1 plus 1 right parenthesis over denominator 2 end fraction close square brackets space space space space space open square brackets therefore 1 plus 2 plus 3 plus... plus straight n space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets
equals Lt with straight h rightwards arrow 0 below straight h open square brackets straight n space straight a plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 end fraction straight h close square brackets space space equals Lt with straight h rightwards arrow 0 below open square brackets straight a space left parenthesis nh right parenthesis space plus space fraction numerator straight n space straight h left parenthesis nh minus straight h right parenthesis over denominator 2 end fraction close square brackets
equals Lt with straight h rightwards arrow 0 below open square brackets straight a space left parenthesis straight b minus straight a right parenthesis space plus space fraction numerator left parenthesis straight b minus straight a right parenthesis space left parenthesis straight b minus straight a minus straight h right parenthesis over denominator 2 end fraction close square brackets
equals space straight a left parenthesis straight b minus straight a right parenthesis space plus space fraction numerator left parenthesis straight b minus straight a right parenthesis space left parenthesis straight b minus straight a minus 0 right parenthesis over denominator 2 end fraction space equals space straight a space left parenthesis straight b minus straight a right parenthesis space plus fraction numerator left parenthesis straight b minus straight a right parenthesis squared over denominator 2 end fraction
equals space left parenthesis straight b minus straight a right parenthesis open square brackets straight a plus fraction numerator 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
therefore space space space space space integral from straight a to straight b of xdx space equals space 1 half left parenthesis straight b squared minus straight a squared right parenthesis
Put a = 1,  b = 2
therefore space space space space space space space space space space space space space integral from 1 to 2 of xdx space equals space 1 half left parenthesis 4 minus 1 right parenthesis space equals space 3 over 2

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


Comparing integral subscript 1 superscript 2 xdx 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 straight x comma space space space straight a space equals space 1 comma space space space straight b space equals space space 2
therefore space space space space straight f left parenthesis straight a right parenthesis space equals space straight a comma space space straight f left parenthesis straight a plus straight h right parenthesis space equals space straight a plus straight h comma space space straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight a plus 2 straight h comma space...... comma space 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 left parenthesis straight n minus 1 right parenthesis space straight h

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

rightwards double arrow  integral subscript straight a superscript straight b xdx space equals space Lt with straight h rightwards arrow 0 below straight h space 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 space straight h right parenthesis right square bracket
                     equals space Lt with straight h rightwards arrow 0 below space straight h space open square brackets left parenthesis straight a plus straight a plus straight a plus.... to space straight n space terms right parenthesis 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 close square brackets
equals space Lt with straight h rightwards arrow 0 below straight h open square brackets straight n space straight a space plus space straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n minus 1 plus 1 right parenthesis over denominator 2 end fraction close square brackets space space open square brackets because 1 plus 2 plus 3 plus.... plus straight n space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets
equals space Lt with straight h rightwards arrow 0 below straight h open square brackets straight n space straight a plus space fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 end fraction straight h close square brackets space equals space Lt with straight h rightwards arrow 0 below open square brackets straight a left parenthesis nh right parenthesis plus fraction numerator nh left parenthesis nh minus 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 space left parenthesis straight b minus straight a right parenthesis plus fraction numerator left parenthesis straight b minus straight a right parenthesis left parenthesis straight b minus straight a minus straight b 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 right parenthesis left parenthesis straight b minus straight a minus 0 right parenthesis over denominator 2 end fraction space 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
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 space equals space left parenthesis straight b minus straight a right parenthesis space open parentheses fraction numerator straight b plus straight a over denominator 2 end fraction close parentheses
therefore space space space space integral subscript straight a superscript straight b straight x space dx space equals space 1 half left parenthesis straight b squared minus straight a squared right parenthesis
space space space space space space space space Put space straight a space equals 1 comma space space space straight b space equals space 2
therefore space space space integral subscript 1 superscript 2 straight x space dx space equals space 1 half left parenthesis 4 minus 1 right parenthesis space equals space 3 over 2
                     

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


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

       space equals space Lt with straight h rightwards arrow 0 below space straight h space left square bracket left parenthesis 1 plus 1 plus 1 plus...... space to space straight n space terms right parenthesis space plus space 2 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
                                                                    plus space straight h squared space open curly brackets 1 squared plus 2 squared plus 3 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 straight h open square brackets straight n plus 2 space straight h space fraction numerator left parenthesis straight n minus 1 right parenthesis straight n over denominator 2 end fraction plus straight h squared space fraction numerator left parenthesis straight n minus 1 right parenthesis space straight n 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 space straight n space straight h space plus space left parenthesis straight n space straight h right parenthesis space left parenthesis straight n space straight h space minus space straight h right parenthesis plus fraction numerator left parenthesis space straight n space straight h right parenthesis space left parenthesis straight n space straight h space minus straight h right parenthesis space left parenthesis 2 space straight n space straight h space minus space straight h right parenthesis over denominator 6 end fraction space close square brackets

equals space Lt with straight h rightwards arrow 0 below open square brackets 2 plus left parenthesis 2 right parenthesis left parenthesis 2 minus straight h right parenthesis plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus straight h right parenthesis space left parenthesis 4 minus straight h right parenthesis over denominator 6 end fraction close square brackets space space space space space space open square brackets because space straight n space straight h space equals space 1 close square brackets
equals space 2 plus left parenthesis 2 right parenthesis space left parenthesis 2 minus 0 right parenthesis space plus fraction numerator left parenthesis 2 right parenthesis space left parenthesis 2 minus 0 right parenthesis space left parenthesis 4 minus 0 right parenthesis over denominator 6 end fraction space equals 2 plus 4 plus 8 over 3 space equals space 26 over 3

       

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