[Multivariate Calc.] Help with triple integrals with cylindrical & spherical coordinates
Hello! I'm studying multivariate calculus and I got along just fine with double integrals and polar coordinates. However, when we have to use cylindrical and spherical coordinates to determine the limits of integration I have trouble. Integrating isn't much problem, getting to write the correct integral in the first place is.
If you have a resource where I can learn more, please link me to it! I've read several chapters from different books and a few youtube videos, but they only do the "easy" stuff. Finding the volume of a sphere, the mass of a cylinder limited by planes, etc.
For example, take this problem from Apostol's Calculus:
·Find the volume of a solid limited by the sphere [; x^2+y^2+z^2 =5 ;] and under the paraboloid [; x^2+y^2=4z ;]
This is the integral that I got:
[; \int_0^{2pi} \int_0^{\sqrt {5}} \int_0^{arctg(2)} \rho^2 \sin \phi\ \mathrm{d\phi d\rho d\theta} ;]
To get to this I found the intersection between the paraboloid and the sphere, z=1, and so x^2+y^2=4. Using spherical coordiantes, this gave me a two equation system with two unknowns, where [; \rho = \sqrt{5} \and\ \tg \phi = 2;] . This is probably where I messed up: I assumed rho started at 0, along with phi.
The solution listed in the book is [; \frac{2}{3} \pi (5 \sqrt{5} - 4);], the one I got was [;\frac{10\pi}{3} (\sqrt{5} - 1);]. Wolfram says I solved that integral correctly, but it wasn't the right integral to use.
More exercises:
·Calculate the mass of the body limited by the sphere centered at the origin with radius sqrt(2) and [; y >= x^2 + z^2;] when the density at each point is proportional to the distance from the point to the y-axis.
I don't really know where to begin. It is a sphere and a paraboloid, like the other one, but since this paraboloid is parallel to the y-axis it makes it difficult to describe in spherical coordinates. Should I start describing it in a coordinate system or should I just find the intersection?
·Calculate the volume of a solid limited by the xy plane, the cylinder [;x^2+y^2=2x;] and the cone [;z=\sqrt{x^2+y^2};]
I'm not sure if I should use spherical or cylindrical for this one. For one, I have a cone which should be described by a fixed phi. On the other hand, I have a cylinder.
I do not expect people to solve these for me, just give me a general guidance, resource, or a guideline to find the limits of integration. That's where I'm really struggling. Sometimes I choose the wrong coordinate system and end up having difficult integrals to do. How do I know what system is the most convenient for me?
Thank you very much!
Posted Jun 9, 2015, 3:12 PM UTC
For the first problem, your integral gives the volume inside the sphere ABOVE a cone which intersects the sphere along the same circle that the paraboloid did. That's a result of all your limits of integration being constants.
For the second problem, you can switch y and z throughout and then use traditional spherical coordinates.
You're right. I calculated the intersection, but forgot to define the lower boundary of the integral. I replaced spherical coordinates on the paraboloid and I got the following lower bound for rho:
[;\rho = 4 \frac{\cos \phi}{\sin^2 \phi};]. However, this makes my integral look like[;\int_0^{2\pi} \int_0^{atan(2)} \int_{4 \frac{\cos \phi}{\sin^2 \phi}}^{\sqrt{5}} \rho^2 \sin\phi\ \mathrm{d\rho d\phi d\theta};], which then I resolved aswhich does not converge. Where did I go wrong?
About the second problem, what do you mean switch y and z? As in change the vector field?
Thank you!
For the first problem, the spherical integral must be split into two, because the paraboloid only serves as a lower bound on rho for the part of the solid inside the upper solid hemisphere. (The paraboloid doesn't sit in the lower solid hemisphere, so the bottom part of the solid is simply the entire lower solid hemisphere.)
Also, your limits on ϕ are wrong because they correspond to the north pole (ϕ=0) down to where the paraboloid and sphere intersect (ϕ=tan^(-1) 2). You need ϕ to start at that intersection and end at the equator for the upper part of the solid.
For the second problem, change the entire problem into
"Calculate the mass of the body limited by the sphere centered at the origin with radius sqrt(2) and z >= x^2 + y^2 when the density at each point is proportional to the distance from the point to the z-axis."
Solve that in spherical using techniques you learned from the first problem. Better yet, use cylindrical.
Absolutely right again. Thanks mate.
My bad, on a rush I mistranslated the exercise. To be precise, it said:
Calculate the volume of a solid with an upper bound a sphere of radius sqrt(5) and lower bound the paraboloid
[;x^2+y^2=4z;]. In essence, it's the region inside the paraboloid up until it intersects the sphere.I was scared of paraboloids and spheres on cylindrical (I had this weird notion that whenever there was a sphere, spherical would be the best option). But I ended up using cylindricals and it got me this nice integral, which gives the correct solution:
[;\int_0^{2\pi} \int_0^2 \int_ {\frac{r^2}{4}}^{\sqrt{5-r^2}} r\ \mathrm{dz\ dr\ d\theta} = \frac{2\pi}{3} (5\sqrt{5}-4);]I think I am starting to get a tiny grasp of this. With more exercises I think I will develop some kind of intuition regarding the subject.
I'm marking this as solved now, since with you and
beerisgood1, I've managed to solve exercises 1) and 3), and 2) is just an extension of the 1) when using the transformation you suggested.Thank you very much!