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Definite integrals

Why a Definite Integral Is Not Always the Total Area

The phrase “area under the curve” can hide an important sign convention. A definite integral counts regions below the horizontal axis negatively, so equal positive and negative regions can cancel even when the geometric area is not zero.

Direct answer

A definite integral gives net signed area; total area adds the magnitudes of regions above and below the axis.

If the integrand changes sign on the interval, split at every relevant zero before calculating total area.

Worked comparison: f(x)=x on [−1,1]

The function x is negative on [−1,0] and positive on [0,1]. Its two triangular regions have equal magnitude but opposite signs. Therefore ∫ from −1 to 1 of x dx equals zero.

Total geometric area cannot cancel. Split at x=0 and reverse the sign of the negative part, or integrate |x|. Each triangle has area 1/2, so the total area is 1.

The same distinction appears in applications. Integrating velocity gives displacement, which can cancel when direction reverses. Integrating speed, or the absolute value of velocity, gives total distance traveled.

Before using a definite-integral result as an area, locate zeros and discontinuities inside the interval. A sign chart decides where absolute values or separate subintervals are required.

1
∫₋₁¹ x dx = 0 (net signed area)
2
∫₋₁⁰ (−x) dx + ∫₀¹ x dx = 1
3
∫₋₁¹ |x| dx = 1 (total area)

Worked examples

∫₋₁¹ x dx
00

The negative and positive signed regions cancel.

Total area between y=x and the axis on [−1,1]
11

Split at zero or integrate |x| so neither region is subtracted.

∫02x−1 dx\int_{0}^{2} x - 1\,\mathrm{d}x
0 net area; 1 total area

The sign change at x=1 separates two equal triangles.

Common mistakes

  • • Reporting a zero definite integral as zero geometric area without checking for cancellation.
  • • Applying one absolute value to the final integral instead of splitting where the function changes sign.
  • • Ignoring an interior discontinuity because the endpoint values look harmless.
  • • Confusing displacement with total distance in a velocity problem.

Try the worked examples in the calculator

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Calculadora de integral definida:xx
Open the calculator without a preset: Calculadora de integral definida

References

Frequently asked questions

Can a definite integral be negative?+

Yes. It is negative when the signed contribution below the axis is larger than the contribution above it.

Is total area always the integral of the absolute value?+

For an integrable function over the stated interval, total area between the graph and the axis is found by integrating its absolute value, usually after splitting at sign changes.

Why do I need to find the zeros first?+

Zeros identify possible sign changes. They tell you where the interval may need to be split before adding geometric areas.

Conclusion

Treat every definite integral as signed until the problem explicitly asks for geometric area. If total area is required, find the sign changes, split the interval and add magnitudes rather than allowing cancellation.

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