Volume of Solid of Revolution: y=x^(-1/2), y=-x^(-1/2), x=1, and x=2; axis y=-2

[kinosh]

The region is bounded by y=x^(-1/2), y=-x^(-1/2), x=1, and x=2. The axis is y=-2.

This problem can be solved by Method of Washers with x-integration
int [1,2] pi[((x^(-1/2)+2)^2)-(-x^(-1/2)+2)^2] dx = 16pi(sqrt(2)-1)

I can’t figure out how to do this by Method of Cylinders with y-integration. I broke it up into 3 separate pieces:
Segment 1: int [-1, -1/sqrt(2)] 2pi(-y^(-2)-1)(-y-1/sqrt(2)+2) dy
Segment 2: int [-1/sqrt(2), 1/sqrt(2)] 2pi(y+2)(1) dy
Segment 3: int [1/sqrt(2),1] 2pi(y^(-2)-1)(y-1/sqrt(2)+2) dy

I solved the bounded regions for x. y=x^(-1/2) becomes x=y^(-2). The problem is with Segment 1 where I get ln y, where y<0, and that’s bad

 

[Dr.Peterson]

The region is bounded by y=x^(-1/2), y=-x^(-1/2), x=1, and x=2. The axis is y=-2.

Segment 1: int [-1, -1/sqrt(2)] 2pi(-y^(-2)-1)(-y-1/sqrt(2)+2) dy
Segment 2: int [-1/sqrt(2), 1/sqrt(2)] 2pi(y+2)(1) dy
Segment 3: int [1/sqrt(2),1] 2pi(y^(-2)-1)(y-1/sqrt(2)+2) dy

I solved the bounded regions for x. y=x^(-1/2) becomes x=y^(-2). The problem is with Segment 1 where I get ln y, where y<0, and that’s bad

Can you explain in detail how you got that integrand? If I'm thinking correctly, there is a sign error in (-y^(-2)-1), and an extra term in (-y-1/sqrt(2)+2). Also, if you still have trouble integrating, please show your steps in that part, so we can make sure you aren't making another little mistake.

While you work on that, I'll finish working on the problem both my way, and trying to integrate what you used. But the more of your own work you show, the easier it is on the rest of us.

 

[mmm4444bot]

int [1,2] pi[((x^(-1/2)+2)^2)-(-x^(-1/2)+2)^2] dx = 16pi(sqrt(2)-1)
…
Segment 1: int [-1, -1/sqrt(2)] 2pi(-y^(-2)-1)(-y-1/sqrt(2)+2) dy
Segment 2: int [-1/sqrt(2), 1/sqrt(2)] 2pi(y+2)(1) dy
Segment 3: int [1/sqrt(2),1] 2pi(y^(-2)-1)(y-1/sqrt(2)+2) dy

My work matches your integration with respect to x. Good job.

Regarding the integration with respect to y, I agree with the Doc, except that I see two sign errors.

This has a sign error.

This has a sign error and an extra term.

This has an extra term.

With corrections, I get 16·Pi·(√2 - 1).

 

[kinosh]

Thanks for the hints. They were very helpful. I need to figure out how to post graphs. That would help explain my work

 

[mmm4444bot]

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