Finding the Volume of the Intersection of Two Cylinders

[TranscendArcu]

Homework Statement

Find the volume of the intersection of the two solid cylinders x² + y² ≤ 1 and y² + z² ≤ 1.

The Attempt at a Solution

Apparently this is done most easily by cartesian coordinates. I have the integral:

$$\int_{-1} ^1 \int_{-sqrt(1-x^2)} ^{sqrt(1-x^2)} \int_{-sqrt(1-y^2)} ^{sqrt(1-y^2)} 1 dzdydx$$But, this is disgusting to integrate (as far as I can tell). I think I either a) have the wrong bounds, or b) have missed a clue to make this problem easier.

 

[Dick]

Changing the order of integration would help a lot. Try integrating dy last instead of dx. Order of integration can make a BIG difference.

 

[TranscendArcu]

So let's see if I can do this:

$$\int_{-1} ^1 \int_{-sqrt(1-y^2)} ^{sqrt(1-y^2)} \int_{-x} ^x 1 dzdxdy$$
$$\int_{-1} ^1 \int_{-sqrt(1-y^2)} ^{sqrt(1-y^2)} 2x dxdy$$This gives an integrand of zero for the last integral. Hmm, did I use incorrect bounds?

 

[Dick]

So let's see if I can do this:

$$\int_{-1} ^1 \int_{-sqrt(1-y^2)} ^{sqrt(1-y^2)} \int_{-x} ^x 1 dzdxdy$$
$$\int_{-1} ^1 \int_{-sqrt(1-y^2)} ^{sqrt(1-y^2)} 2x dxdy$$This gives an integrand of zero for the last integral. Hmm, did I use incorrect bounds?

Umm, yes, you used the wrong bounds. Shouldn't the z bounds be determined by y^2+z^2<=1?? Why would you think they should be -x to x?

 

[TranscendArcu]

I have two possible problems with those bounds: first, and without doing any actual calculations, those bounds don't make the integral look very much nicer than what I had in #1; second, I would be concerned that we "lose" information about x by not including it somewhere in the bounds (that is, I see we have z's relationship to y and x's relationship to y, but not x's relationship to z). I used -x≤z≤x because that's what I got by setting,

x^2 +y^2 = y^2 + z^2, which suggests,

z = +/- x.

 

[Dick]

I have two possible problems with those bounds: first, and without doing any actual calculations, those bounds don't make the integral look very much nicer than what I had in #1; second, I would be concerned that we "lose" information about x by not including it somewhere in the bounds (that is, I see we have z's relationship to y and x's relationship to y, but not x's relationship to z). I used -x≤z≤x because that's what I got by setting,

x^2 +y^2 = y^2 + z^2, which suggests,

z = +/- x.

I have some problems with that. How does x^2+y^2<=1 and y^2+z^2<=1 imply that x^2+y^2=y^2+z^2?? I suggested integrating dy last exactly because then the x bounds depend only on y and the z bounds depend only on y. Why do think there is some other dependency?

 

[TranscendArcu]

Okay. I think I understand this now. I did the calculation out and got 16/3, which I have marked as the right answer. I guess doing the integration in that order was easier.

 

[Dick]

Okay. I think I understand this now. I did the calculation out and got 16/3, which I have marked as the right answer. I guess doing the integration in that order was easier.

A LOT easier. And yes, it is 16/3.