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Integrals

Question
CBSEENMA12032350
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Evaluate integral subscript 1 superscript 4 left parenthesis straight x squared minus straight x right parenthesis space dx  as the limit of a sum.

Solution

Comparing integral subscript 1 superscript 4 left parenthesis straight x squared minus straight x right parenthesis dx 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 squared minus straight x comma space space straight a space equals space 1 comma space space space space straight b space equals space 4
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                  ..................................................................
           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 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 right parenthesis squared minus left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis
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 space straight h right parenthesis right square bracket
       therefore space space space integral subscript 1 superscript 4 left parenthesis straight x squared minus straight x right parenthesis dx space equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket 0 plus open curly brackets left parenthesis 1 plus straight h right parenthesis squared minus left parenthesis 1 plus straight h right parenthesis close curly brackets space plus space open curly brackets left parenthesis 1 plus 2 straight h right parenthesis squared minus left parenthesis 1 plus 2 straight h right parenthesis close curly brackets plus.....
                                                                                   plus open curly brackets left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis squared minus left parenthesis 1 plus stack straight n minus 1 with bar on top space straight h right parenthesis close curly brackets right square bracket
                 equals space Lt with straight h rightwards arrow 0 below space straight h left square bracket open curly brackets straight n plus 2 straight h space left parenthesis 1 plus 2 plus 3 plus... plus stack straight n minus 1 with bar on top right parenthesis plus straight h squared left parenthesis 1 squared plus 2 squared plus.... plus left parenthesis straight n minus 1 right parenthesis squared close curly brackets
                                                                               negative open curly brackets straight n plus left parenthesis 1 plus 2 plus.... plus stack straight n minus 1 with bar on top right parenthesis space straight h close curly brackets right square bracket

            WiredFaculty
                                                                                            open square brackets because space space straight n space straight h space equals space straight b space minus straight a space equals space 4 space minus space 1 space equals space 3 close square brackets
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