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find surface area obtained by rotating `y=sqrt(t), t in[4,9]`  about the t-axis

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pensivegal | (Level 1) Honors

Posted September 20, 2013 at 12:02 AM via web

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find surface area obtained by rotating `y=sqrt(t), t in[4,9]`  about the t-axis

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted September 20, 2013 at 5:11 AM (Answer #1)

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The surface area of the solid obtained by rotating the curve `y = sqrt t` about the t-axis for t in[4, 9] is given by the integral `int_4^9 2*pi*y*ds` where `ds = sqrt(1 +(dy/dt)^2) dt`

`y = sqrt t`

`(dy)/(dt) = 1/(2*sqrt t)`

`ds` = `sqrt(1 +(1/(2*sqrt t))^2) dt`

=> `ds = sqrt((4t + 1)/(4t)) dt`

`int_4^9 2*pi*y*ds`

= `int_4^9 2*pi*y*sqrt((4t + 1)/(4t)) dt`

= `int_4^9 2*pi*sqrt t*sqrt((4t + 1)/(4t)) dt`

= `pi*int_4^9 sqrt(4t + 1) dt`

= `(pi/6)*[(4t+1)^(3/2)]_4^9`

= `(pi/6)*(37^(3/2) - 17^(3/2))`

`~~ 81.14`

The required surface area is 81.14

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