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Integrals

Question
CBSEENMA12032425
Wired Faculty App

If integral subscript straight a superscript straight b straight x cubed dx space equals space 0 space and space if space integral subscript straight a superscript straight b straight x squared dx space equals space 2 over 3. find both a and b.

Solution
integral subscript straight a superscript straight b space straight x cubed space dx space equals space 0 space space space rightwards double arrow space space open square brackets straight x to the power of 4 over 4 close square brackets subscript straight a superscript straight b space equals space 0
rightwards double arrow space space space space 1 fourth left parenthesis straight b to the power of 4 minus straight a to the power of 4 right parenthesis space space equals space 0 space rightwards double arrow space straight b to the power of 4 minus straight a to the power of 4 space equals space 0 rightwards double arrow space space space space space left parenthesis straight b squared minus straight a squared right parenthesis space left parenthesis straight b squared plus straight a squared right parenthesis space equals space 0 space space rightwards double arrow space space straight b squared minus straight a squared space equals space 0 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 b squared plus straight a squared space not equal to 0 close square brackets rightwards double arrow space space space space space left parenthesis straight b minus straight a right parenthesis space left parenthesis straight b plus straight a right parenthesis space equals space 0 space space rightwards double arrow space space space straight b plus straight a space equals space 0 space space space space space space space open square brackets because space straight b not equal to straight a space as space integral subscript straight a superscript straight b straight x squared dx not equal to 0 close square brackets therefore space space space space straight b space equals space minus straight a 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 space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis
   Also      integral subscript straight a superscript straight b straight x squared dx space equals space 2 over 3 space space space space space space space space rightwards double arrow space space space space space space space open square brackets straight x cubed over 3 close square brackets subscript straight a superscript straight b space equals space 2 over 3

rightwards double arrow     1 third left parenthesis straight b cubed minus straight a cubed right parenthesis space equals space 2 over 3         rightwards double arrow       straight b cubed minus straight a cubed space equals space 2
 
therefore      left parenthesis negative straight a right parenthesis cubed minus straight a cubed space equals space 2                                             open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets
 
therefore        negative space 2 space straight a cubed space equals space 2                   rightwards double arrow    straight a cubed equals negative 1      rightwards double arrow     a = -1
therefore  we have a = -1,   b = 1

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