Volume of solid x^2 + (y-1)^2 =1 about y-axis

[BethW86]

Homework Statement

Hello, I am to find the volume of the solid given by x² + (y-1)²=1 rotated about the y-axis. I may use either shells or cylindrical method. I attempted shell method, but am just learning this, still foggy and this is the one question that isn't coming out right.

Homework Equations

My understanding is that with shells, my rectangles are placed parallel to the rotation axis?

The Attempt at a Solution

I took the equation and solved for y, so that y=+- sqrt(1-x²)+1
I set up my integral as pi INT[-1,1] x(sqrt(1-x²)+1) dx
I'm not sure my integration is correct: pi [1/2sqrt91-x^2)x +x+1/2sin^-1(x)] |-1 to 1
When I did this, I got (pi^2)/4. The books says I should get (4pi)/3
Thanks so much for any direction. Feel free to use simple terms, I'm new at this :)

 

[Mark44]

Homework Statement

Hello, I am to find the volume of the solid given by x² + (y-1)²=1 rotated about the y-axis. I may use either shells or cylindrical method. I attempted shell method, but am just learning this, still foggy and this is the one question that isn't coming out right.

Homework Equations

My understanding is that with shells, my rectangles are placed parallel to the rotation axis?

The way you have things set up, it looks like you are using disks, not shells.
For each disk, the volume is $\Delta V = \pi (radius)^2 \Delta x$.
Here, the radius of a typical disk is the y-value that you show below, minus 1. The radius squared would then be 1 - x². This leads to an integral that is simpler than what you show below, plus, it gives the correct answer, always a good thing!

The Attempt at a Solution

I took the equation and solved for y, so that y=+- sqrt(1-x²)+1
I set up my integral as pi INT[-1,1] x(sqrt(1-x²)+1) dx

You can exploit the symmetry of the rotated object (a sphere) and take twice the volume as x ranges between 0 and 1.

I'm not sure my integration is correct: pi [1/2sqrt91-x^2)x +x+1/2sin^-1(x)] |-1 to 1
When I did this, I got (pi^2)/4. The books says I should get (4pi)/3
Thanks so much for any direction. Feel free to use simple terms, I'm new at this :)

The book's answer is correct. The volume of a sphere of radius 1 is $4\pi/3$.

 

[BethW86]

Thanks so much, that helped!

 

[Mark44]

When I do problems where something is being rotated around an axis or other line, I draw two sketches: one of the region that is being rotated, and another of the solid that results. In the latter drawing, I add a sketch of the typical volume element. I then calculate the volume or area or whatever of this element, which pretty much gives me my integrand.