Disk Washer Method: Vol of Solid Generated by y=1/sqrt(x)

[Justabeginner]

Homework Statement

Find the volume of the solid generated by revolving the region bounded by y=1/sqrt(x) and y=0 for 1≤x≤2 about the line y=-1

Homework Equations

Disk Method? ∏ * ∫[f(x)]^2 dx

The Attempt at a Solution

A and B (lower and upper limits will be 1 and 2)?
∏ * ∫(1/sqrt(x)^2 dx
∏ * ∫1/x dx
∏ * ln x
Since limits are 2 and 1, my final answer is: ∏ * ln 2

Is this right? Thanks.

 

[Mark44]

Homework Statement

Find the volume of the solid generated by revolving the region bounded by y=1/sqrt(x) and y=0 for 1≤x≤2 about the line y=-1

Homework Equations

Disk Method? ∏ * ∫[f(x)]^2 dx

This formula is applicable only if the rotation is around the x-axis.

The formula would be better written as ∏ * ∫[radius]²dx

The Attempt at a Solution

A and B (lower and upper limits will be 1 and 2)?
∏ * ∫(1/sqrt(x)^2 dx
∏ * ∫1/x dx
∏ * ln x
Since limits are 2 and 1, my final answer is: ∏ * ln 2

Is this right? Thanks.

No. Your integrand doesn't reflect the fact that the region is being rotated around the line y = -1. You can't use a disk in this case - you need to use a washer if you use vertical strips, or you could use shells if you go with horizontal strips.

 

[Justabeginner]

Would the correct way to set it up then be

∏ * ∫([1/sqrt(x)] + 1)^2) ?

And follow up with:
∏* ∫[1/x + 2/sqrt(x) + 1]
∏ * ([ln x] + 4sqrt(x) + x) ?

Thank you so much for all your help.

 

[Mark44]

Would the correct way to set it up then be

∏ * ∫([1/sqrt(x)] + 1)^2) ?

No, this isn't set up correctly. You're using disks when you should be using washers.

For a washer, the volume is $\pi [(R_{outer})^2 - (R_{inner})^2]Δx$.

And follow up with:
∏* ∫[1/x + 2/sqrt(x) + 1]
∏ * ([ln x] + 4sqrt(x) + x) ?

Thank you so much for all your help.

 

[Justabeginner]

But I just don't understand, even after drawing this diagram, how there's an inner AND an outer radius? If you could explain that to me, I'd appreciate it.

 

[Mark44]

The region to be rotated is roughly trapezoidal in shape. Its bottom edge is the x-axis, the left and right edges are the lines x = 1 and x = 2, and the top edge is the curve y = 1/√x.

When the region is rotated around the line y = -1, you get a kind of ring-shaped solid that is hollow. The inner radius is 1, and the outer radius extends from the line y = -1 to the curve.

 

[Justabeginner]

Thank you! I am horrible at visualizing things in 3D and your description made it so much clearer.

But how would I determine the length of the outer radius? I know that the points on the curve are (1, 1) and (2, sqrt(2)/2)- how would that help me in this problem, or would it not be of any use?

 

[Mark44]

Points on the curve are of the form (x, 1/√x). The upper end of the outer radius is at this point, and the lower end is at (x, -1). How far apart are these points?

 

[Justabeginner]

They are (1/sqrt(x) + 1) units apart?

 

[Mark44]

Yes, so this is the outer radius. Don't forget, it needs to be squared.

 

[Justabeginner]

So what I get now is:
∏ *∫{[(1/sqrt(x) + 1)^2] - (1^2)} * Δx (Isn't delta x = 1 since x=1 and x=2 are bounding the region?)
∏ * ∫(1/x) + 2/sqrt(x) + 1 - (1)
∏ * ∫(1/x) + 2/sqrt(x)
∏ * (ln x + 4sqrt(x))
And since x=1 and x=2 are the left and right bounds, I'd plug that into solve for the final volume?

∏* [(ln 2 + 4*sqrt(2) - (ln 1 + 4 sqrt(1)]
∏ * [ln 2 + 4sqrt(2) - 4]

Is this correct?

 

[Mark44]

So what I get now is:
∏ *∫{[(1/sqrt(x) + 1)^2] - (1^2)} * Δx (Isn't delta x = 1 since x=1 and x=2 are bounding the region?)

No, that's not what Δx means. It corresponds loosely with dx in the integral.

I don't have time to look at this, right now, but will later on if nobody else jumps in.

∏ * ∫(1/x) + 2/sqrt(x) + 1 - (1)
∏ * ∫(1/x) + 2/sqrt(x)
∏ * (ln x + 4sqrt(x))
And since x=1 and x=2 are the left and right bounds, I'd plug that into solve for the final volume?

∏* [(ln 2 + 4*sqrt(2) - (ln 1 + 4 sqrt(1)]
∏ * [ln 2 + 4sqrt(2) - 4]

Is this correct?

 

[Justabeginner]

Okay thank you so much! (Yes I just realized, delta x can also be used for dx- I don't know why I even said that -_-)

 

[Mark44]

So what I get now is:
∏ *∫{[(1/sqrt(x) + 1)^2] - (1^2)} * Δx (Isn't delta x = 1 since x=1 and x=2 are bounding the region?)
∏ * ∫(1/x) + 2/sqrt(x) + 1 - (1)
∏ * ∫(1/x) + 2/sqrt(x)
∏ * (ln x + 4sqrt(x))
And since x=1 and x=2 are the left and right bounds, I'd plug that into solve for the final volume?

∏* [(ln 2 + 4*sqrt(2) - (ln 1 + 4 sqrt(1)]
∏ * [ln 2 + 4sqrt(2) - 4]

Is this correct?

Yes, that's what I get, too.

 

[Justabeginner]

Thank you very much sir!