What is the area of the sky that is covered by the Hubble Ultra Deep Field image? I have found two different numbers for the area of the sky covered by the Hubble Ultra Deep Field (HUDF). According to this, the image is roughly 2.4 arcminutes wide. The image is also attached to the link with a scale showing it is around 2.4 arcminutes by 2.4 arcminutes. 
I might be wrong, but I think this means that the area covered is 2.4 $\times$ 2.4 = 5.76 squared arcminutes.
But according to this, the image covers 11 square minute-of-arc region in the 
southern sky.
I actually doubt the method I used to calculate the area of 5.76 squared arcminutes. So am I wrong ? and if not, then which is the correct number ?
 A: Your method is correct.  When the angle is "small" so that we can ignore curvature, then the rectangular solid angle is just the product of the two side angles.   (This doesn't hold for "large" angles).
I cannot find a perfect explanation, but  one source of this confusion may be that the UDF was imaged by two separate instruments, one optical, one infrared.
This page shows some details, including the fact that the optical survey is 3 arcminutes square and the IR is 2.4 arcminutes square.  So I think the entire UDF is 11 square arcminutes (3.4 arcminutes square), but the image released does not cover the entire area.  
An additional piece I found that supports this size is that the Hubble ACS wide-field channel (found here) is 202 arcseconds on a side, and this page (which has an image showing the relative size of the moon, the HUDF and some other Hubble surveys) states the HUDF is exactly the size of one wide-field image.  202 arcseconds squared is 11.3 square arcminutes.
A: It's complicated. There wasn't just one Hubble Ultra Deep Field (UDF) observation, but several, taken at different times, with different instruments, but pointed in (almost) the same direction.
The first image you refer to was taken with the WFC Infra red camera in 2009, in a near infrared band (1-1.6 microns). The area covered by this camera is $2.4$ arminutes on a side. i.e. 5.76 square arcminutes.
The second Beckwith et al. (2005) paper you refer to discusses a UDF taken with the Advanced Camera for Surveys (ACS) in 2004, an instrument that works in the visible part of the spectrum, edging just into the near infrared (300-900nm). The image is approximately square because it had to be taken using different roll angles of the spacecraft and the image was dithered. The final (cropped) square image is 11 square arcminutes.
There is then a 2012 Hubble UDF taken again with the WFC IR instrument, in the same direction, with increased exposure times, especially in the 1.05 micron band, and which added a 1.4 micron deep image. This also covers 5.76 square arcminutes. It also added further exposure (another 128 orbits!) to the 800nm optical/near IR exposure with the ACS instrument (11 square arcminutes). http://adsabs.harvard.edu/abs/2013ApJS..209....3K
EDIT: The number of square degrees on the sky is $4\pi (180/\pi)^2 =41253$ square degrees, which gives your number. This is just a solid angle of $4\pi$ steradians (or square radians) converted into degrees.
