Signed area and net change
An integral adds area above the axis and subtracts area below it. So it gives net change, which can cancel out to zero even when the true area does not.
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Analogy
Walking 5 km east then 5 km west: you travelled 10 km but your net displacement is zero.
Example
Integral from 0 to pi is +2.Integral from pi to 2 pi is -2.Net integral = 0.True area = 2 + 2 = 4.
In practice
Care is needed when a question asks for total distance or total area, not net change.
Questions people ask
How do I find the true area when a curve crosses the axis?
Find where the curve crosses the x-axis, split the integral there, and take the absolute value of each piece before adding. For sine from 0 to 2 pi, the pieces are +2 and -2, so the true area is 2 + 2 = 4.
What does net change mean in practice?
It is the overall difference between the start and the end. If velocity is positive then negative, the integral of velocity gives displacement, which can cancel out. The integral of speed, which is never negative, gives total distance travelled. Always read the question to see which one it wants.