How to find the volume when solids intersect?

[Adesh]

Homework Statement

Find the volume when three cylinders of radius 1 intersect orthogonally.

Relevant Equations

Volume = area x height

I know that to find the volume under a surface and above a boundary we have to integrate twice. I can explain myself with an example :-
Lets' consider that we need to find the volume under the surface $z = \sqrt{1-x^2}$ and above the region bounded by $y^2 = x$ and positive x-axis and $x=5$ . So, what I would do is to first define the limits of x and y , so $-\sqrt{x} < y < \sqrt{x}$ and for x we have $0 < x < 5$ so we can now carry out our integrals like this $\int_ {-\sqrt{x}}^{\sqrt{x}} \sqrt{1-x^2} dy$ which I can call $A(x)$ and then I can integrate $A(x)$ from $0 to 5$ and hence I will get the desired volume.

But when it comes to three intersecting cylinders I can't figure out under what and above what I have to find the volume . I have added an attachment which shows the solid that we get after intersection, but I still can not figure out how to go on? Can anybody please help to indemnify the upper and lower surfaces? Is it that my knowledge (that dividing the lower surface into small vanishing rectangles and calculating the volume of rectangular columns and then summing it up) is inadequate for this problem?

I must confess that there are lot of solutions available for the Computation of Volume under three Intersecting Cylinders on google search but it's also true that all of them do it by symmetry which also is not understandable by me due to lack of visualization . I resp[ect this site therefore I have asked here.

THANK YOU, ANYHELP HELP WILL BE MUCH APPRECIATED.

 

[Charles Link]

This is a rather well known problem of the Steinmetz Solid. See https://www.physicsforums.com/threads/math-challenge-by-charles-link-1.907510/#post-5716117 $\\$ When I solved the problem, I worked with one corner, and recognized there is, in addition, a symmetry about $y=x$, with the bore in the y-direction being lower than the bore in the x-direction, on the bottom side for $y<x$. Thereby, you work with half of a corner, and multiply the answer by 16. $\\$ Note: This is a slight variation of the question you were asked, but the one can be obtained from the other. It helps to draw a Venn diagram to sort out the different volumes. In your case, I believe you are asked to find the volume that includes all 3 cylinders. To do that, it might help to first find the volume common to two cylinders, etc. $\\$ For a Venn diagram, like the one you need for this problem, see https://www.quora.com/3-identical-c...es-What-is-the-ratio-of-the-intersection-area We are working with volumes here, but the regions representing the types of intersections of the volumes are analogous. In this problem, I believe they are asking you to compute the volume represented by $A$ in this diagram.

 

[Charles Link]

I probably need to add some to the above, so you can follow the methodology. It is somewhat difficult to compute the overlap of even two cylinders by trying to look at the limits of integration of the part of the volume that is common to both. Instead, the analogous problem can be worked of drilling into a unit cube, and computing the volume that remains after the holes are bored. Let's first work with two cylinders in the x and y directions: If there is no volume common to both, then $V_{remaining}=1-2 (\frac{\pi}{4})$. If there is a volume common to both $V_{common}$, then $V_{remaining}=1-2 (\frac{\pi}{4})+V_{common}$. The problem then is to find $V_{remaining}$ in order to compute $V_{common}$. The limits of integration are simpler for $V_{remaining}$. It should be apparent that you have 8 identical corners of the cube, each with the same volume. In addition, there is a symmetry about $y=x$. Suggestion is to first work this two cylinder case. The 3 cylinder one is only slightly more difficult, and methods of finding the overlapping regions are similar. You also need the answer to this two cylinder case for reasons that will become apparent when you solve the Venn diagrams for the different regions in the 3 cylinder case. $\\$ Note: In the above, I'm taking the diameter as one unit, rather than the radius.

 

[Adesh]

I probably need to add some to the above, so you can follow the methodology. It is somewhat difficult to compute the overlap of even two cylinders by trying to look at the limits of integration of the part of the volume that is common to both. Instead, the analogous problem can be worked of drilling into a unit cube, and computing the volume that remains after the holes are bored. Let's first work with two cylinders in the x and y directions: If there is no volume common to both, then $V_{remaining}=1-2 (\frac{\pi}{4})$. If there is a volume common to both $V_{common}$, then $V_{remaining}=1-2 (\frac{\pi}{4})+V_{common}$. The problem then is to find $V_{remaining}$ in order to compute $V_{common}$. The limits of integration are simpler for $V_{remaining}$. It should be apparent that you have 8 identical corners of the cube, each with the same volume. In addition, there is a symmetry about $y=x$. Suggestion is to first work this two cylinder case. The 3 cylinder one is only slightly more difficult, and methods of finding the overlapping regions are similar. You also need the answer to this two cylinder case for reasons that will become apparent when you solve the Venn diagrams for the different regions in the 3 cylinder case. $\\$ Note: In the above, I'm taking the diameter as one unit, rather than the radius.

I followed your hint that there is a symmetry by $y =x$. The figure which I have attached is the top view of intersection of three cylinders $x^2 + y^2 = 1$ , $x^2 + y^2 = 1$ $y^2 + z^2 = 1$ and in my figure purple cylinder is $y^2 + z^2 =1$ and the green one is $x^2 + z^2 =1$

the blue lines which are scarcely visible are $y=x$ and $y=-x$. So, following your advice I did this :-
In region the surface above ( if we look from ordinary xy plane) is $z = \sqrt{1-x^2}$ and the region below over which $z= \sqrt{1-x^2}$ lies is defined by $y= -1$ $y=x$ and $y=-x$. Therefore we need go find the volume above a region which is $D = {(x,y) : -1<y<0 , y < x < -y }$ and now we can go for integral that is $\int_{y}^{-y} \sqrt{1-x^2} dx$ and then we have to integrate this with respect to y from $-1$ to $0$ which will yield an answer of (3$\pi$ - 4)/6. Now, region 2 will have the same volume.

What I should do next? Am I correct this far? I want to thank you for giving me your valuable time and attention.

 

[Charles Link]

I find it easier to write the cylinders as $(x-\frac{1}{2})^2+(z-\frac{1}{2})^2=(\frac{1}{2})^2$ and $(y-\frac{1}{2})^2+(z-\frac{1}{2})^2=(\frac{1}{2})^2$. You then find $V_{remaining}=16 \int\limits_{0}^{1/2}\int\limits_{0}^{x} (\frac{1}{2}-\sqrt{\frac{1}{4}-(x-\frac{1}{2})^2}) \, dy \,dx$. I need to double-check the last double integral, but I think I got it right. Trigonometric substitution methods work for evaluating the integral. Note that the dy is simple, because there is no y dependence in the integrand. Notice also that we only needed to use the equation for the y-bore here.

 

[Charles Link]

And yes, I worked through the above integral again, (for the two-cylinder case), and evaluated it=it gets the correct answer. The substitution $u=\frac{1}{2}-x$ comes in handy, and then it comes in two parts. One part requires the trigonometric substitution $u=\frac{\cos{\theta}}{2}$, while the other part has a simple closed form. I will be glad to give additional details if need be. $\\$ Putting it a bore/cylinder in the z-direction for the 3 cylinder problem for $V_{remaining}$ makes for y limits that are from $0$ to $x$ for only part of the interval of dx from $0$ to $1/2$. For part of the interval, the y limits are determined by the circle $(x-\frac{1}{2})^2+(y-\frac{1}{2})^2=(\frac{1}{2})^2$. In any case, the 3 cylinder problem, with bores into a cube, is readily workable as well. You can probably google the integrals, or use Wolfram, etc., but it is a good exercise to try to evaluate them on your own. :)

 

[Adesh]

Sir can you please explain that drilling method. How to go about drilling?

 

[Charles Link]

Sir can you please explain that drilling method. How to go about drilling?

See the link in post 2 of the Math Challenge we did about a year or two ago (March 2017). A couple of the participants posted graphics that show the concept of the drilling into the cube. $\\$ It is kind of a complementary type problem: Finding the remaining volume after drilling, as opposed to finding the volume that is common to all 3 cylinders. The latter can be found by evaluating the numbers from the former process.$\\$ For the 3 cylinders, there are 3 types of regions that get drilled: "A" common to all 3 cylinders ;"B" common to two of the cylinders, and "C" containing just one cylinder but not the other two. For 3 cylinders, $V_{remaining3}=1-3V_C-3V_B-V_A$ as you should be able to see with a Venn diagram. In addition $\frac{\pi}{4}=V_C+2V_B+V_A$ as you can also see from the diagram. In addition, for just two cylinders, $V_{common2}=V_B+V_A$. $\\$ I linked to the type of Venn diagram you need, also in post 2, but will explain in more detail if the process is unclear. (There is only one region A, but there are 3 B regions, each with volume $V_B$, and 3 C type regions, each with volume $V_C$). The various overlapping regions of the rings each represent the different volumes when the 3 cylinders overlap. (You of course aren't computing an overlapping area here). You need to compute $V_{remaining}$ for the two and three cylinder cases, in order to work the equations, and ultimately get $V_A$, which is the volume common to all 3 cylinders. $\\$ One additional equation you need that I will repeat here (see post 3 above) is $V_{remaining2}=1-2(\frac{\pi}{4})+V_{common2}$.

 

[Charles Link]

And to summarize: In the method I used, I computed $V_{remaining2}$, and $V_{remaining3}$, with integrals that are computed when two and three cylinders are drilled out of a cube. From these integrals, (and they took some effort to evaluate), $V_A$ was ultimately computed. Venn diagrams of rings were helpful in illustrating the different types of volumes that occur with the overlapping cylinders. In the calculation for $V_A$, there are basically 3 equations, and 3 unknowns: $V_A, V_B,$ and $V_C$.