The transverse energy (E-T) in Pb-Pb collisions at 2.76 TeV nucleon-nucleon center-of-mass energy (root s(NN)) has been measured over a broad range of pseudorapidity (eta) and collision centrality by using the CMS detector at the LHC. The transverse energy density per unit pseudorapidity (dE(T)/d eta) increases faster with collision energy than the charged particle multiplicity. This implies that the mean energy per particle is increasing with collision energy. At all pseudorapidities, the transverse energy per participating nucleon increases with the centrality of the collision. The ratio of transverse energy per unit pseudorapidity in peripheral to central collisions varies significantly as the pseudorapidity increases from eta = 0 to vertical bar eta vertical bar = 5.0. For the 5% most central collisions, the energy density per unit volume is estimated to be about 14 GeV/fm(3) at a time of 1 fm/c after the collision. This is about 100 times larger than normal nuclear matter density and a factor of 2.6 times higher than the energy density reported at root s(NN) = 200 GeV at the Relativistic Heavy Ion Collider.