Results 1 to 3 of 3

Math Help - surface area and integration

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    91

    surface area and integration

    Calculate the area of the portion E+ of the ellipsoid that lies above the xy plane by making use of the parameterisation  r_{\to} (\theta, \phi)= (2cos \phi sin \theta , 2 sin \phi cos \theta , cos \theta ).

    Hint:  \int \sqrt {a^2 + b^2 u^2} du =  \frac {1}{2} \sqrt {a^2 + b^2 u^2} + \frac {a^2} {2b} log(bu+\sqrt {a^2 + b^2 u^2} )

    I know that I need to evaluate  A=\int \int_{E+} || r_{\theta} \times r_{\phi} || for  0 \leq \theta \leq \pi/2 and  0 \leq \phi \leq 2\pi

    Now, I have found  || r_{\theta} \times r_{\phi} || =  2sin \theta \sqrt{(sin \theta)^2 + 4(cos \theta)^2} but am stuck on how to use the hint?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Aug 2009
    Posts
    143
    Quote Originally Posted by nmatthies1 View Post
    Calculate the area of the portion E+ of the ellipsoid that lies above the xy plane by making use of the parameterisation  r_{\to} (\theta, \phi)= (2cos \phi sin \theta , 2 sin \phi cos \theta , cos \theta ).

    Hint:  \int \sqrt {a^2 + b^2 u^2} du =  \frac {1}{2} \sqrt {a^2 + b^2 u^2} + \frac {a^2} {2b} log(bu+\sqrt {a^2 + b^2 u^2} )

    I know that I need to evaluate  A=\int \int_{E+} || r_{\theta} \times r_{\phi} || for  0 \leq \theta \leq \pi/2 and  0 \leq \phi \leq 2\pi

    Now, I have found  || r_{\theta} \times r_{\phi} || =  2sin \theta \sqrt{(sin \theta)^2 + 4(cos \theta)^2} but am stuck on how to use the hint?
    Seems you want to calculate

    \int || r_{\theta} \times r_{\phi} || \; d\theta =\int 2\sin \theta \sqrt{\sin^2 \theta + 4\cos^2 \theta} \; d\theta=-2\int \sqrt{1 + 3\cos^2 \theta} \; d \cos \theta = -2\int \sqrt{1 + 3u^2} \; du (Let  u=\cos \theta)

    Then use the hint to proceed ....
    Last edited by mr fantastic; September 18th 2009 at 07:53 AM. Reason: Restored original post
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,417
    Thanks
    1330
    Quote Originally Posted by nmatthies1 View Post
    Calculate the area of the portion E+ of the ellipsoid that lies above the xy plane by making use of the parameterisation  r_{\to} (\theta, \phi)= (2cos \phi sin \theta , 2 sin \phi cos \theta , cos \theta ).
    This does NOT give an ellipsoid. \vec{r}(\theta, \phi)= (2cos\phi sin\theta, 2 sin\phi sin\theta, cos\theta) gives the ellipsoid \frac{x^2}{4}+ \frac{y^2}{4}+ z^2= 1.

    Hint:  \int \sqrt {a^2 + b^2 u^2} du =  \frac {1}{2} \sqrt {a^2 + b^2 u^2} + \frac {a^2} {2b} log(bu+\sqrt {a^2 + b^2 u^2} ) .

    I know that I need to evaluate  A=\int \int_{E+} || r_{\theta} \times r_{\phi} || for  0 \leq \theta \leq \pi/2 and  0 \leq \phi \leq 2\pi

    Now, I have found  || r_{\theta} \times r_{\phi} || =  2sin \theta \sqrt{(sin \theta)^2 + 4(cos \theta)^2} but am stuck on how to use the hint?
    In the form I have given, \phi is the "longitude" and ranges from 0 to 2\pi, and \theta is the "co-latitude" and ranges from 0 to \pi.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: September 12th 2011, 08:22 PM
  2. Surface area integration (flux integral)
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 24th 2011, 03:00 PM
  3. Calculate the surface area of the surface
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 26th 2009, 04:03 AM
  4. Volume, Surface Area, and Lateral Surface Area
    Posted in the Geometry Forum
    Replies: 1
    Last Post: April 14th 2008, 11:40 PM
  5. [SOLVED] [SOLVED] Surface Area using integration
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 24th 2008, 05:55 AM

Search Tags


/mathhelpforum @mathhelpforum