Getting an specific frequency component from sampled audio
The commonly used Goertzel filter (or algorithm) used to detect a single frequency component is essentially a computational optimization of computing 1 bin of a DFT this way.  

Note that a both a Goertzel filter and 1 bin of an un-windowed (or rectangular windowed) DFT don't just respond to a single frequency, but have a non-zero bandwidth filter response in the shape of a Sinc function, with most of the main lobe response on the order of Fs/N wide.  So the 32 Hz bin of your half-second-long FFT will contain most of a 32.45 Hz component, as well as vice versa with the Goertzel result.

Note that if the k is fractional, or not integer, which corresponds to a Goertzel filter length that is not an exact integer multiple of the period of the frequency in question, there will be some dependency of the magnitude result on the phase of the frequency component with respect to some point in the sampling window (say, the start).

And, unlike with a full FFT, 1 bin of a DFT won't tell you whether the measured component is significant compared to adjacent frequencies (is a peak), or to the rest of the signal (is above the noise floor).

But otherwise, yes.