A-level · Beta

Integration and signed area

Move the limits, compare signed areas and refine a trapezium approximation.

Drag A and B along the x-axis to change the limits. Green and red show the signs of f(x).

∫−12(−1+x2) dx=0\int_{-1}^{2}(-1+x^{2})\,dx=0
Drag the labelled rings or use their arrow keys.-4-224-6-336xy0AB

Tab to a handle, then use the arrow keys. Use the sliders for precise values.

Areas below the x-axis subtract from areas above it.

Function
f(x)=−1+x2f(x)=-1+x^{2}
Signed integral
00
Total geometric area
≈2.6667\approx 2.6667
Trapezium estimate
≈0.125\approx 0.125
Approximation error
≈0.125\approx 0.125

The shading shows the sign of f(x). The integral is signed accumulation; it is not automatically the total geometric area. Decimal estimates are rounded to 4 places.

Predict, test and explain

Step 1 of 3

Predict

What is the signed integral of x from −1 to 1?

InvestigateCan the signed integral be zero while there is visible area?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

What is the signed integral of x from −1 to 1?

∫−11x dx\int_{-1}^{1}x\,dx

Hint

Find an antiderivative, then evaluate upper limit minus lower limit.

Reveal answer

The equal positive and negative contributions cancel; there is still geometric area.

00

Build an explanation

Hint 1

Find an antiderivative, then evaluate upper limit minus lower limit.

Hint 2

Below-axis contributions have negative sign.

Hint 3

Reversing the limits reverses the sign; equal limits give zero.

Worked example

This example uses fixed values, separate from the diagram controls.

Evaluate the integral of x² from 0 to 3.

  1. An antiderivative is x³/3.
  2. At x = 3 this gives 27/3 = 9.
  3. Subtract its value at x = 0, which is 0.

Answer: 9.

Connect this idea

Watch out: A signed integral is not always total area. Equal regions below and above the axis cancel.

Specification and learning route

AQA H1, H2, H3 · Edexcel Pure 8.1, Pure 8.2, Pure 8.3

AS core; A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Power-rule differentiation; Algebra and indices; Area and signed graphs.

AQA specification · Edexcel specification

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