Results 1 to 12 of 12

Math Help - Need Help with vector calculus problems!

  1. #1
    Newbie
    Joined
    Jul 2008
    Posts
    11

    Need Help with vector calculus problems!

    1) Use the triple integral to find the volume of the solid bounded above and below by the sphere x^2 + y^2 + z^2 = 9 and cylinder x^2 + y^2 = 4

    2) Find the work done by a force F(x,y,z) = (x+y)i + (xy)j - (z^2)k acting on a particle that moves along the line segment from (0,0,0) to (1,3,1) and then along the line segment from (1,3,1) to (2,-1,4)

    3) Let S be the first-octant portion of the paraboloid z= x^2 + y^2 that is cut off the by the plane z=4. If F(x,y,z) = (x^2 + z)i + (zy^2)j + (x^2 + y^2 + z)k, find the flux of F through S.

    If you could help me with any of these problems much would be appreciated. If you could show your work then that would be great too! Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by davidson89 View Post
    1) Use the triple integral to find the volume of the solid bounded above and below by the sphere x^2 + y^2 + z^2 = 9 and cylinder x^2 + y^2 = 4

    [snip]
    1) Integrate in the order dz dx dy.

    z-integral terminals: Lower terminal is z = - sqrt{9 - x^2 - y^2} (lower surface). Upper terminal is z = sqrt{9 - x^2 - y^2} (upper surface).

    The double integral dx dy is over the region of the xy-plane defined by x^2 + y^2 = 4. I'm sure you can set up the integral terminals. Personally, I'd switch to polar coordinates once the z-integration has been done .....

    Note: You can use the symmetry of the problem .... Two times the triple integral where the lower terminal is z = 0 (lower surface) and the upper terminal is z = sqrt{9 - x^2 - y^2} (upper surface) ......
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by davidson89 View Post
    [snip]

    2) Find the work done by a force F(x,y,z) = (x+y)i + (xy)j - (z^2)k acting on a particle that moves along the line segment from (0,0,0) to (1,3,1) and then along the line segment from (1,3,1) to (2,-1,4)
    [snip]
    2) You need to evaluate the line integral int F.dr over the two line segments.

    Line segment from (0, 0, 0) to (1, 3, 1):

    The parametric equation of the line segment is x = t, y = 3t, z = t where t = 0 to t = 1.

    So the line integral becomes

    int[t = 0, t = 1] [(t + 3t)i + (3t^2)j - (t^2)k].[1i + 3j + 1k] dt

    (you know where this expression has come from, right?)


    Line segment from (1,3,1) to (2,-1,4):

    Get the parametric equation and proceed in a similar way to the above.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by davidson89 View Post
    [snip]

    3) Let S be the first-octant portion of the paraboloid z= x^2 + y^2 that is cut off the by the plane z=4. If F(x,y,z) = (x^2 + z)i + (zy^2)j + (x^2 + y^2 + z)k, find the flux of F through S.

    If you could help me with any of these problems much would be appreciated. If you could show your work then that would be great too! Thanks!
    More definitions. Where are you stuck here? Have you drawn a picture? Can you set up the flux integral? Can you do the integrations?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jul 2008
    Posts
    11
    Hey, thanks for your help. Yes i am having trouble setting up the integrals for 1 and 3. I cant seem to figure out the conditions for the integrals. Also, i really do not have an idea to the flux problem, your clarification would be much appreciated on how to do them
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by davidson89 View Post
    Hey, thanks for your help. Yes i am having trouble setting up the integrals for 1 and 3. I cant seem to figure out the conditions for the integrals. Also, i really do not have an idea to the flux problem, your clarification would be much appreciated on how to do them
    1) V = int int int dx dy dz and I've already given the integral terminals for this triple integral as well as advice on how to approach the integrations.

    --------------------------------------------------------------------------------------------------------------

    For 3), you must at least have the definition Flux = int int F . dS in your notes or textbook .....

    The surface is the paraboloid z= x^2 + y^2. Use this equation to get an expression for vector dS in terms of dx dx (your notes and/or textbook should have the necessary formula for doing this):

    vector dS = (.........) dx dy

    where you get to fill in the dotted line.

    Now do the dot product F.(.......). This converts the flux integral into a routine double integral in the xy-plane.

    The region of integration in the xy-plane is the circle x^2 + y^2 = 4 (substitute z = 4 into z = x^2 + y^2). This suggests converting to polar coordinates (for an easier calculation) to evaluate the double integral.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jul 2008
    Posts
    11
    hey so for number 2 would the integration be:

    integral from 0 to 1 [ (1+t+3-4t) + (1+t)(3-4t)(-4) - (1+3t)^2(3)]

    i get the answer of -139/6

    If i add this with the result of the first integration, i get -37/2 is this correct
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by davidson89 View Post
    hey so for number 2 would the integration be:

    integral from 0 to 1 [ (1+t+3-4t) + (1+t)(3-4t)(-4) - (1+3t)^2(3)]

    i get the answer of -139/6

    If i add this with the result of the first integration, i get -37/2 is this correct
    The set up of the line integral along the second line segment looks fine. I haven't bothered to check the value.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Jul 2008
    Posts
    11
    So i dont get for number 3 how to set up the integral.... if it is in the order dx dy dz, then the integral terminals are still
    z = 0 to z = (4 - x)/2
    x = 0 to x = 4
    y = 0 to y = 2. ?

    So it would be

    int(0 to 2) int(0 to 4) int(0 to 4-x/2) [ dx dy dz ]

    or should it be

    int(0 to 4-x/2) int (0 to 2) int (0 to 4) [ dx dy dz]

    could it be done like this?

    int (0 to 4) int (0 to 2) int(0 to (4-x)/2) [ dz dy dx]

    thanks for your clarification
    Last edited by davidson89; July 28th 2008 at 03:57 PM.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by davidson89 View Post
    [snip]
    3) Let S be the first-octant portion of the paraboloid z= x^2 + y^2 that is cut off the by the plane z=4. If F(x,y,z) = (x^2 + z)i + (zy^2)j + (x^2 + y^2 + z)k, find the flux of F through S.

    If you could help me with any of these problems much would be appreciated. If you could show your work then that would be great too! Thanks!
    Quote Originally Posted by davidson89 View Post
    So i dont get for number 3 how to set up the integral.... if it is in the order dx dy dz, then the integral terminals are still
    z = 0 to z = (4 - x)/2
    x = 0 to x = 4
    y = 0 to y = 2. ?

    So it would be

    int(0 to 2) int(0 to 4) int(0 to 4-x/2) [ dx dy dz ]

    or should it be

    int(0 to 4-x/2) int (0 to 2) int (0 to 4) [ dx dy dz]

    could it be done like this?

    int (0 to 4) int (0 to 2) int(0 to (4-x)/2) [ dz dy dx]

    thanks for your clarification
    I don't see how this relates to Q3 as posted. I have discussed Q3 in post #6. What part of that discussion don't you understand.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie
    Joined
    Jul 2008
    Posts
    11
    I am sorry i meant to post this reply in the other post. Haha My question was how to set up the integral for this problem

    Let S be the surface of the region bounded by the coordinate planes and the planes x + 2z = 4 and y = 2. Use the Divergence Theorem to the flux of F(x,y,z) = (2xz)i + (xyz)j + (yz)k through S
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by davidson89 View Post
    I am sorry i meant to post this reply in the other post. Haha My question was how to set up the integral for this problem

    Let S be the surface of the region bounded by the coordinate planes and the planes x + 2z = 4 and y = 2. Use the Divergence Theorem to the flux of F(x,y,z) = (2xz)i + (xyz)j + (yz)k through S
    What part of my reply (#3) at this thread: http://www.mathhelpforum.com/math-he...-calculus.html

    do you not understand regarding this question.


    Seriously, I'm struggling to keep a track of all your questions. It's just getting too confusing for me. In the future:

    1. Present one question per thread. Not several.

    2. When a reply is given, ask follow-up questions at that thread if you still don't understand. If you post the same question again in other places, people just end up saying the same things several times which wastes their time.

    Go back. Read the replies I have given to each of your questions. Then attempt to understand how to do the question. If you're still stuck after doing these things, state clearly where you're stuck.

    Many of your problesm look like they could be resolved if you went back and revised the relevant mathematics in your textbook and/or class notes.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Vector Calculus (Position Vector)
    Posted in the Calculus Forum
    Replies: 6
    Last Post: August 23rd 2011, 02:43 PM
  2. Vector Calculus
    Posted in the Calculus Forum
    Replies: 10
    Last Post: March 22nd 2011, 02:30 PM
  3. Replies: 2
    Last Post: February 22nd 2011, 05:19 AM
  4. Replies: 2
    Last Post: June 25th 2010, 11:41 PM
  5. vector calculus - vector feilds
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 25th 2010, 02:17 AM

Search Tags


/mathhelpforum @mathhelpforum