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Consider the integral of sin x over the interval from 0 to 2pi.
In this interval the value of sin x rises from 0 to 1 then falls through 0 to -1 and then rises again to 0. In other words the part of the sin x function between 0 and pi is 'above' the axis and the part between pi and 2pi is 'below' the axis.
The value of this integral is zero because although the areas enclosed by the parts of the function between 0 and pi and pi and 2pi are the same the integral of the latter part is negative.
The point I am trying to make is that a definite integral gives the area between a function and the horizontal axis but areas below the axis are negative.
The integral of sin x over the interval from 0 to pi is 2.
The integral of six x over the interval from pi to 2pi is -2.
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