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Integrals

Question
CBSEENMA12032349
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Evaluate  integral subscript 1 superscript 3 straight x squared space dx as the limit of a sum.

Solution

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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