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can anyone solve the volume ques plz??
y = x^(1/2) ln x
Volume = pie ( y )^2
Volume = pie [ x^(1/2) ln x ]^2
Volume = pie [x (ln x)^2]
Now use integration by parts to integrate 'x (ln x)^2'
x (ln x)^2
(x^2/2)(ln x)^2 - [x^2 /2 * 2 ln x * 1/x]
(x^2/2)(ln x)^2 - [x ln x]
Now integrate 'x ln x'
x ln x
(x^2/2) ln x - [x^2/2 * 1 /x]
(x^2/2) ln x - [x/2 ]
(x^2/2) ln x - x^2/4
(x^2/2)(ln x)^2 - [x ln x]
(x^2/2)(ln x)^2 - [ (x^2/2) ln x - x^2/4 ]
(x^2/2)(ln x)^2 - (x^2/2) ln x + x^2/4
Now put the limits 'e' (upper) and '1' (lower).
Volume = pie [(e^2/2)(ln e)^2 - (e^2/2) ln e + e^2/4] - [(1^2/2)(ln 1)^2 - (1^2/2) ln 1 + 1^2/4 ]
Volume = pie [(e^2/2) - (e^2/2) + e^2/4] - [ 1/4 ]
Volume = pie [e^2/4] - [ 1/4 ]
Volume = (pie/4)(e^2 - 1)
Therefore, the volume is '(pie/4)(e^2 - 1)'.