  Understanding the magnetic excitations in high-transition temperature
(high-$T_c$) copper oxides is important because they may mediate the electron
pairing for superconductivity. By determining the wavevector ({\bf Q}) and
energy ($\hbar\omega$) dependence of the magnetic excitations, one can
calculate the change in the exchange energy available to the superconducting
condensation energy. For the high-$T_c$ superconductor YBa$_2$Cu$_3$O$_{6+x}$,
the most prominent feature in the magnetic excitations is the resonance.
Although the resonance has been suggested to contribute a major part of the
superconducting condensation, the accuracy of such an estimation has been in
doubt because the resonance is only a small portion of the total magnetic
scattering. Here we report an extensive mapping of magnetic excitations for
YBa$_2$Cu$_3$O$_{6.95}$ ($T_c\approx 93$ K). Using the absolute intensity
measurements of the full spectra, we estimate the change in the magnetic
exchange energy between the normal and superconducting states and find it to be
about 15 times larger than the superconducting condensation energy. Our results
thus indicate that the change in the magnetic exchange energy is large enough
to provide the driving force for high-$T_c$ superconductivity in
YBa$_2$Cu$_3$O$_{6.95}$.
