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because integrals are just the sum of all the rectangles that start at y=0 and go "up" to the curve, "up" is in quotes because if the curve is negative then "up" means start at y=0 and go down, they still are rectangles like any other but have negative area (area can perfectly be negative, don't let physicist nonsense tell you otherwise) so that gets substracted to the total instead of added

>>1509860
>Shouldn't it give you the area below the curve to infinity?
Then the area would be -infinity anywhere the curve is negative, and that would be utterly useless.
Plus, the derivative at those points would be 0, which doesn't make sense when the original curve was non-zero.
The integral is the anti-derivative, so you have to get back the original curve if you differentiate the integral.

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>>1509860
Because you can simply move the curve upwards by whatever T amount that ensures it's always positive, then calculate it's integral and then subtract the rectangle of (length of interval)*T to get the same original curve

>>1509860
It's just distance to the x axis
"up" and "down" don't have any meaning here

>>1509872
>>1509890
>>1509897
>>1509970
I think I get it now. Integration is the sum of values. In the case of functions those values are f(x). It's not "just" the area under a curve, it's the area between the curve and the x-axis.

>>1510146
It's the area under the curve, but under means upward if you go negative.
imagine that you are tasked to build several buildings, some with a "negative height" (meaning a hole in the ground), how much material do you use?
The answer is the amount of material you used above ground minus the one you got from the hole

>>1510146
>it's the area between the curve and the x-axis.
Correct