Which possible methods can be used to determine if a certain part of the brain is active? The question goes back to this tweet:

Julia‏ @JuliaHass
I just learned that elephants think humans are cute the way humans think puppies are cute (the same part of the brain lights up when they see us) so pack it in, nothing else this pure and good is happening today.

The skeptics stack exchange post about the tweet refers to elephants, but I'm interested in a general answer.
PET (positron emission tomography) will measure glucose consumption, e.g. https://www.ncbi.nlm.nih.gov/pubmed/2147116
fMRI will measure changes in blood flow, or more precisely measuring the blood-oxygen-level dependent (BOLD) response. https://www.nature.com/scitable/blog/brain-metrics/what_does_fmri_measure


Note that I'm being a bit simplistic here, as with PET you can measure a lot of different things (like activation of circuits depending on specific neurotransmitters) depending on the tracer used. Here's a table from the book PET and SPECT in Neurology:



The downside is that PET has lower spatial resolution than fMRI and substantially lower temporal resolution, meaning chnages over time are hard to pick up with PET. If you care about quick changes, then MEG or EEG are the fastest, but you have solve a difficult inverse problem to figure out where the changes came from, which requires making quite a few assumptions to get a (unique) solution.

And let me add (for infotainment purposes) "[f]MRI Killed The Radiotracer".