What is the relationship between the integral of a function and the area under the graph of that function? Not a homework question.
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They are they’re almost the same thing. For simple functions that are positive for all values in the range, they are equal. It gets a bit weird when you have something like a sine function or cosine, where the function can be negative.
The Integral of sin(x) from 0 to pi is:
[-cos(x) – – > – cos(pi) – (- cos(0)) – – > – (-1) – (-1) = 2]
but the Integral from zero to 2pi (one complete period) is:
[ – cos(2pi) – (-cos(0)) – – > -(1) – (-1) = 0]
Intuitively, if you look at the graph of sin(x) from 0 to 2pi you think “well it’s symmetrical so if the first half area is 2 then the total area is 4“ and you’d be correct, but the Integral is 0 because the second half of the function has a ‘negative area’ (not a real thing) according to integration rules.
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