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.

abkapseln87 2022-09-30 Answered
Need help with the following question about finding volume of improper integral
Find 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 x
How would I go about solving this because I am unsure how to integrate that integral.
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Answers (1)

bequejatz8d
Answered 2022-10-01 Author has 6 answers
Step 1
I 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 2
Since lim x 1 ln x = 0, we get
π [ 1 ln x ] 2 = π ( 0 ( 1 ln 2 ) ) = π ln 2
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