Need help with the following question about finding volume of improper integralFind the exact volume of the solid created by rotating the region bounded by f ( x ) = 1 x ln ⁡ x and the x-axis on the interval [ 2 , ∞ ). State the method of integral used.My issue specifically is that I have no clue how to integrate the integral because I am using the disk method and the integral ends up being ∫ 2 ∞ π ⋅ 1 x ( ln ⁡ ( x ) ) 2 d xHow would I go about solving this because I am unsure how to integrate that integral.

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Need help with the following question about finding volume of improper integralFind the exact volume of the solid created by rotating the region bounded by   f  (  x  )  =      1                  x            ln      ⁡      x       and the x-axis on the interval   [  2  ,      ∞    ). State the method of integral used.My issue specifically is that I have no clue how to integrate the integral because I am using the disk method and the integral ends up being      ∫    2    ∞    π  ⋅      1          x      (      ln      ⁡      (      x      )              )        2                d  xHow would I go about solving this because I am unsure how to integrate that integral.

bequejatz8d · Accepted Answer

Step 1I see   x  (  ln  ⁡  x      )    2   in the denominator, which suggests that the antiderivative involves       1          ln      ⁡      x      . And indeed:      ∫    2    ∞    π  ⋅      1          x      (      ln      ⁡      (      x      )              )        2                d  x  =  π            [      −              1                  ln          ⁡          x                    ]        2    ∞  Step 2Since       lim          x      →      ∞            1          ln      ⁡      x        =  0, we get  π            [      −              1                  ln          ⁡          x                    ]        2    ∞    =  π      (    0    −          (      −              1                  ln          ⁡          2                    )        )    =      π          ln      ⁡      2

Find the exact volume of the solid created by rotating the region bounded by f(x)=1/(sqrt(x) ln x) and the x-axis on the interval [2, infty). State the method of integral used.

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