Signed Integral vs Total Area: Split the Curve at Its Zeros

A definite integral adds signed contributions above and below the axis. Total geometric area is ∫|f(x)|dx, so split at sign changes and make each area contribution nonnegative.

The same curve can answer two different questions

If f stays nonnegative on an interval, its definite integral equals the area between the graph and the horizontal axis. If f becomes negative, its integral subtracts those below-axis regions. Positive and negative contributions can cancel even when the total geometric area is substantial.

Find the curve’s zeros inside the interval and inspect the sign between them. A zero where the function only touches the axis may not change sign. For continuous polynomial examples, one test point in each interval settles the sign.

Worked example with three regions

Let f(x) = x² − 1 on [−2, 2]. Its zeros are −1 and 1. The function is positive on [−2, −1) and (1, 2], and negative between −1 and 1. An antiderivative is F(x) = x³/3 − x.

The signed integral is F(2) − F(−2) = 2/3 − (−2/3) = 4/3. For geometric area, calculate the outer regions and reverse the central negative contribution. Each outer region has integral 4/3, while the middle signed integral is −4/3.

The total area is 4/3 + 4/3 + 4/3 = 4 square units. The signed answer 4/3 and area answer 4 are both correct for their respective questions. Reporting the absolute value of the final signed integral would give only 4/3 and would still be wrong for total area.

Why one final absolute value is insufficient

Cancellation happens before you take an absolute value of the final total. To preserve every region’s size, apply the sign correction separately. For y = x on [−1, 1], the signed integral is zero, but two right triangles each have area 1/2, so total area is 1.

In a rate context, a velocity integral gives displacement, while integrating speed gives distance travelled. That is the same signed-versus-nonnegative distinction, with units of length rather than geometric square units.

Common mistake

Do not add the same arbitrary constant repeatedly when evaluating definite integrals. Use one antiderivative and subtract endpoint values on each subinterval. Also label whether you are reporting a signed integral, displacement, distance or geometric area.

A quick sketch can support the sign table, but exact split points should come from solving f(x) = 0 where possible. Approximate roots introduce approximation into the area calculation and should be identified as such.

Area mein har region positive; signed integral mein cancellation ho sakta hai.

Three practice questions with worked answers

1. For f(x) = x on [−1, 1], find the signed integral and total area.

   The signed integral is [x²/2]₋₁¹ = 0. The two triangular regions each have area 1/2, so total area is 1.

2. For f(x) = −2 on [0, 3], find both quantities.

   The signed integral is −2 × 3 = −6. The geometric area is the nonnegative rectangle area 2 × 3 = 6.

3. For f(x) = x − 1 on [0, 2], find total area.

   The sign changes at 1. Each side forms a triangle of base 1 and height 1, so total area is 1/2 + 1/2 = 1. The signed integral is 0.

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References and authorship

Original examples and practice questions, prepared with AI assistance and editorial arithmetic review. Related definitions and methods: OpenStax: The Definite Integral. No textbook or exam-board affiliation is claimed.