Signed area and integral bookkeeping

Try

Sketch y=x from x=−1 to x=2. One triangle lies below the axis and another above it. If you want net signed accumulation, subtract the first triangle’s area. If you want all the geometric area between the line and axis, add both magnitudes. Calculate both results before introducing any antiderivative.

Predict

Predict whether an integral of zero means a function is zero everywhere. Then predict what happens when the limits are reversed. Reversing an oriented interval is different from changing the sign of a physical area; an integral records orientation as well as magnitude.

Experiment

Use the following signed-area ledger for the line y=x.

Interval	Geometric area	Signed contribution
[−1,0]	1/2	−1/2
[0,2]	2	2
Combined	5/2	3/2

Now reverse the full integral: from 2 to −1 the signed value is −3/2. The geometric region has not changed. Draw arrows beside the x-axis to distinguish orientation from the curve’s position above or below it.

Understand

For integrable functions, an integral over a zero-width interval is zero. Reversing bounds negates the value. Splitting at a point c gives ∫ₐᵇf=∫ₐᶜf+∫ᶜᵇf, with oriented conventions even when the endpoints are not listed in increasing order. These rules allow a long accumulation to be assembled from or compared with shorter ones.

Linearity means ∫(αf+βg)=α∫f+β∫g for constants α and β over the same bounds. It follows from distributing each finite Riemann sum before taking limits. If ∫f=3 and ∫g=−1 on one interval, then ∫(2f−3g)=6+3=9. The coefficients must be constants; replacing one by a changing function creates a product, for which no such separation is generally valid.

For example, on [0,1], taking f=g=x gives ∫fg=∫x²=1/3, whereas (∫f)(∫g)=1/4. Integrals do not distribute across multiplication. Nor is ∫f/g usually a quotient of integrals. Check a claimed identity on a simple known example before using it in a larger calculation.

If a≤b and f(x)≤g(x) throughout the interval, comparison gives ∫f≤∫g. The pointwise assumption concerns every input, not merely measured samples. If 2≤f≤5 on [1,4], comparison with constant functions bounds its integral between 6 and 15. Reversing the bounds reverses the signs and associated order, so the interval orientation must remain visible.

For geometric area between a graph and the axis, integrate |f| or split at sign changes. Cancellation may be meaningful for signed quantities such as inflow minus outflow, but it hides the total amount of activity. A continuous function with zero integral can be nonzero almost everywhere: f(x)=x on [−1,1] provides a concrete example.

Apply

Use the practice problems to distinguish information that follows from a known integral from information that requires the entire function. Explain whether your answer is signed accumulation, nonnegative geometric area, or a bound. Counterexamples should satisfy the proposed identity’s stated assumptions rather than changing the problem.

Continue

Next let the upper endpoint move. The resulting accumulated quantity is itself a function, and the first fundamental theorem connects its derivative to the current rate.