Integrals

  • Question 1
    CBSEENMA12032343

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

    Solution

    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

    Question 2
    CBSEENMA12032344

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

    Solution

    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.

    Question 3
    CBSEENMA12032345

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



    Solution

    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
                         

    Question 4
    CBSEENMA12032346

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


    Solution

    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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