  We demonstrate a high-Q ($>5 \times 10^{6}$) swept-frequency membrane
mechanical resonator achieving octave resonance tuning via an integrated heater
and an unprecedented acceleration noise floor below 1 $\mu$g Hz$^{-1/2}$ for
frequencies above 50 kHz. This device is compatible with established batch
fabrication techniques, and its optical readout is compatible with
low-coherence light sources (e.g., a monochromatic light-emitting diode). The
device can also be mechanically stabilized (or driven) with the same light
source via bolometric optomechanics, and we demonstrate laser cooling from room
temperature to 10 K. Finally, this method of frequency tuning is well-suited to
fundamental studies of mechanical dissipation; in particular, we recover the
dissipation spectra of many modes, identifying material damping and coupling to
substrate resonances as the dominant loss mechanisms.
