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Integrals

Question
CBSEENMA12032349

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
           WiredFaculty
WiredFaculty
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
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