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`xy=3 , y =1, y=4 , x=5` Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the...

Let's use the method of disc for evaluating the volume of the solid generated.


As per the method of discs `V=intAdx`  or `V=intAdy`  , where A stands for Area of a typical disc , `A=pir^2`


 and`r=f(x)`  or `r=f(y)` depending on the axis of revolution.


Given `xy=3 , y=1 , y=4 , x=5`


and the region is rotated about the line x=5


Consider a disc perpendicular to the line of revolution,


Then the radius of the disc will be `(5-x)`


Since  `xy=3, x=3/y`


Radius of the disc = `(5-3/y)`


`V=int_1^4pi(5-3/y)^2dy`


`V=piint_1^4(25-2(5)(3/y)+(3/y)^2)dy`


`V=piint_1^4(25-30/y+9/y^2)dy`


`V=pi[25y-30ln(y)+9(y^(-2+1)/(-2+1))]_1^4`


`V=pi[25y-30ln(y)-9/y]_1^4`


`V=pi{[25(4)-30ln(4)-9/4]-[25(1)-30ln(1)-9/1]}`


`V=pi(100-9/4-30ln(4)-25+9)`


`V=pi(84-9/4-30ln(4))`


`V=pi(327/4-30ln(4))`


`V~~126.17`

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