  X-ray cavities and extended radio sources (`cocoons') surrounding active
galactic nuclei (AGN) have been detected by the Chandra X-ray mission and radio
interferometers. A joint analysis of X-ray and radio maps suggests that
pressure values of non-thermal radio-emitting particles derived from the radio
maps are not sufficient to inflate the X-ray cavities. We propose using the
Sunyaev-Zel'dovich (SZ) effect, whose intensity strongly depends on the
pressure, to find the hitherto undetected, dynamically-dominant component in
the radio cocoons.
  We demonstrate that the spectral function at a frequency of 217 GHz has an
absolute maximum at a temperature higher than $10^9$ K, therefore the
measurement of the SZ effect at this frequency is a powerful tool for
potentially revealing the dynamically-dominant component inside AGN jet-driven
radio cocoons. A new method is proposed for excluding the contribution from the
low energy, non-relativistic electrons to the SZ effect by means of
observations at two frequencies. We show how one may correct for a possible
contribution from the kinematic SZ effect. The intensity maps of the SZ effect
are calculated for the self-similar Sedov solution, and application of a
predicted ring-like structure on the SZ map at a frequency of 217 GHz is
proposed to determine the energy released during the active jet stage. The SZ
intensity map for an AGN cocoon in a distant elliptical is calculated using a
2-D numerical simulation and including relativistic corrections to the SZ
effect. We show the intensity spectrum of the SZ effect is flat at high
frequencies if gas temperature is as high as $k_\mathrm{b} T_{\mathrm{e}}=500$
keV.
