Question

Find the surface area generated when y= x^3/12 + 1/x from x=1 to x=2 is rotated around the x-axis

Answer 1

Find the surface area generated from x = 1 to x = 2:

`y=(x^3)/(12)+(1)/(x)`

The definite integral of the above function from x = 1 to x = 2, will be used to generate the surface area

`\begin{gathered} \int ^2_1(x^3)/(12)+(1)/(x)dx \\ \mathrm{Apply\: the\: Sum\: Rule}\colon\quad \int f\mleft(x\mright)\pm g\mleft(x\mright)dx=\int f\mleft(x\mright)dx\pm\int g\mleft(x\mright)dx \end{gathered}`
`\begin{gathered} =\int ^2_1(x^3)/(12)dx+\int ^2_1(1)/(x)dx \\ =\int ^2_1\lbrack(x^4)/(4(12))+\ln x\rbrack \end{gathered}`
`\begin{gathered} =\int ^2_1\lbrack(x^4)/(4(12))+\ln x\rbrack \\ =\int ^2_1\lbrack(x^4)/(4(12))+\int ^2_1\ln x\rbrack \\ =((2^4)/(48)-(1^4)/(48))+(\ln 2-\ln 1) \\ =((16)/(48)-(1)/(48))+\ln 2-0 \\ =(15)/(48)+\ln (2) \\ (5)/(16)+\ln (2) \end{gathered}`

Hecne the surface area of the function = 5/16 + In(2)

`(5)/(16)+\ln (2)`