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to measure what?
I know they're used to determine the area between the curve of a function and a the x-axis, but what does this area represent?
Really integrals are used to measure areas and distances. When you use them in physics you will see their true power, which is quite powerful.
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It can mean a bunch of things. It could be literal area if you were to have the graph model the outer edges of something. If you take a graph of something like:
You can revolve that graph into a 3d figure:
There, now you have the volume of something, whatever you need to calculate. In physics, you could use an integral to measure impulses of collisions or something. You use integration to undo differentiation to help you figure out how something changed or what it changed from. I havent done TOO many applications where you could see a ton of real life stuff, but it can really be used to do a lot, and it definitely is not always limited to area.