Rotating the region bounded by y=x^3, y=0, x=2 around the line y=8.

Makenna Booker 2022-07-21 Answered
Finding volume of solid using shell method.
Rotating the region bounded by y = x 3 , y = 0 , x = 2 around the line y = 8.
I just want to double check if my initial formula is on the right track.
V y = 0 8 ( 8 y ) ( y 1 3 ) d y
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Answers (2)

Ali Harper
Answered 2022-07-22 Author has 16 answers
Explanation:
You are on the right track, but I think you want
V = 0 8 2 π r ( y ) h ( y ) d y = 0 8 2 π ( 8 y ) ( 2 y 1 / 3 ) d y

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termegolz6
Answered 2022-07-23 Author has 3 answers
Step 1
Since you are rotating around the line y = 8, which is parallel to the x-axis, it makes sense to integrate along the x-axis (i.e., integrate dx) rather than along y. The outer radius at the point x (let's call it R(x)) will be from y = 8 to y = 0 and the inner radius at the point x (let's call it r(x)) will be from y = 8 to y = 0 and the inner radius at the point x (let's call it r(x)) will be from y = 8 to y = x 3 ; this all happens over the region where x goes from 0 to 2.
Step 2
0 2 π [ R ( x ) 2 r ( x ) 2 ] d x = 0 2 π [ ( 8 0 ) 2 ( 8 x 3 ) 2 ] d x =

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