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Q: Why is the integral/antiderivative the area under a function?

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Physicist: If you’ve taken calculus, then at some point you learned that to find the area under a function (generally written ) you need to find the anti-derivative of that function. In what follows “f” is a function, and “F” is its anti-derivative (that is: F’ = f). Intuitively: Say you’ve got a function f(x), and the area under f(x) (up to some value x) is given by A(x). So if the height of the function (which is just the function) is the rate at which the area changes, then f is the derivative of the area: A’=f. You can just as easily write this as or (since F’ =f).