Re: Integrals In Spherical Coordinates

From: meyousikmann (meyousikmann_at_nospamyahoo.com)
Date: 12/04/04 

Date: Sat, 4 Dec 2004 16:47:41 -0600

"Daniel Grubb" <grubb@lola.math.niu.edu> wrote in message
news:cnt3u3$hgk$1@news.math.niu.edu...
>
>>This is homework so I don't want the answer....just some hints.
>
>
>>Let W be the solid bounded above by x^2+y^2+z^2=9 and below by phi=pi/3.
>>Calculate the mass of W if the density at each point is directly
>>proportional to the distance above the xy plane.
>
>
>>As I have it drawn, it is an upward cone about the origin with a slanted
>>plane of z = -x - y +3 for a top. I am having a little difficulty setting
>>up the triple integral as well as the density function. Here is what I
>>have
>>so far:
>
> First of all, you should go back and do some algebra. I suspect that
> you solved for z by going from z^2=9-x^2-y^2 and taking square roots
> to get z=3-x-y. If that is the case, I would almost immediately give
> you 0 points for this problem. Nobody in third semester claculus
> should do this mistake *ever*.
>

Well, let me first say thanks for the pointers....they are truly
appreciated. Second, may I also say that I am glad I do not have you for an
instructor. I realized my mistake after walking away from the problem and
coming back with a "fresh" perspective. Don't you think zero points is a
little harsh? That is typically why students fear math and science
courses....no room for error. We learn from mistakes. You scare us away
with zero points. How about a bit of encouragement to keep plugging instead
of a blanket statement that makes the student want to quit. It is called
learning.

>>2 pi pi/6 ???
>> / / /
>> | | | ??? p^2 sin phi d-rho d-phi d-theta
>>/ / /
>>0 0 0
>
>>Anyone got any pointers on this one?
>
> Well, your order is messed up. To do any multiple integral,
> figure out the inside limits *first*. The reasoning goes as follows:
>
> If phi and theta are fixed, what are the allowed values of rho
> for the solid? You can think of the rho values as sweeping out
> a little radial line. These rho values can depend on both phi and theta.
> The smallest value is the lower limit on the inside integral, and the
> largest the upper limit.
>
> Now, let phi vary but keep theta fixed. As phi varies, the little radial
> line sweeps out a planar cross-section of your volume. What
> range of phi values are allowed in your figure? These values may depend
> on theta *but not rho* and give the limits of the middle integral.
>
> Finally, what range of theta values are needed for the planar cross
> sections to sweep out the whole volume? These values won't depend on
> either rho or phi and give the limits for the outside integral.
>
> --Dan Grubb

Relevant Pages

  • Re: Integrals In Spherical Coordinates
    ... If phi and theta are fixed, what are the allowed values of rho ... These rho values can depend on both phi and theta. ... on theta *but not rho* and give the limits of the middle integral. ...
    (sci.math)
  • Re: Integration in polar coordinates?
    ... No, assuming you are talking about spherical coordinates (rho, theta, ... phi) here, there is a sinfactor in addition to the r^2, which you ...
    (sci.math)
  • 3D Polar plot
    ... Now I want to plot the data as function of theta and phi, with rho -now- representing the data/measured value. ...
    (comp.soft-sys.matlab)
  • Re: Have equation of motion, what is the Lagrangian?
    ... rho = phi ... This is the Euler Lagrange equation of a Lagrangian theory if you can ... (rho delta Psi),t. ... symplectic structure of the field theory. ...
    (sci.physics)
  • Re: Another newbie question
    ... r and theta. ... > will be able to manipulate the object with some attributes but not ... class's usefulness to have rho and theta read-only (i.e., ... def fromPolar: ...
    (comp.lang.python)