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The definite integral gives the signed area, counting regions below the axis as negative. Integrating the absolute value instead gives the total geometric area, which is what a paint or material estimate needs.
Signed area
A = ∫ₐᵇ f(x) dx
Total area
A = ∫ₐᵇ |f(x)| dx
x³/3 gives 9, and since the curve is above the axis the signed and unsigned areas agree.
Whenever the curve crosses the axis. For sin x from 0 to 2π the signed area is 0 but the total area is 4.