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You have to integrate. There is usually no easy shortcut, except for some very simple functions.

When finding the area between to functions subtract the bottom function from the bottom function. Then integrate that from the starting to ending points. (If the functions switch which is on top and which is on the bottom over the interval you're integrating over, you'll have to split the problem into smaller problems around that point and add the areas together.)

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Q: What is the easiest way to find area between two functions?
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