CalculusAdditional / Further Maths

Signed area and definite integrals

Distinguish signed integrals from total geometric area.

  • A definite integral gives signed area: area above the horizontal axis is positive and area below it is negative.
  • Geometric area is always non-negative. Equal areas on opposite sides can cancel in an integral.
  • For total area, split at the roots and add the magnitudes of the separate integrals.

Example 1

Find the signed integral and the total shaded area .

The shaded region between the labelled curves and the stated limits.
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to 0, below the axis.Integrate from 0 to 2, above the axis.The signed contributions can cancel.Add the two positive areas.

Example 2

Find the signed integral and the total shaded area .

The shaded region between the labelled curves and the stated limits.
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to -1, below the axis.Integrate from -1 to 3, above the axis.The signed contributions can cancel.Add the two positive areas.

Your turn

Question 1

Find the signed integral and the total shaded area .

The shaded region between the labelled curves and the stated limits.
Check answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to -1, below the axis.Integrate from -1 to 0, above the axis.The signed contributions can cancel.Add the two positive areas.

Area is measured in square units.

Question 2

Find the signed integral and the total shaded area .

The shaded region between the labelled curves and the stated limits.
Check answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to 1, below the axis.Integrate from 1 to 4, above the axis.The signed contributions can cancel.Add the two positive areas.

Area is measured in square units.

Question 3

Find the signed integral and the total shaded area .

The shaded region between the labelled curves and the stated limits.
Check answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to 0, below the axis.Integrate from 0 to 5, above the axis.The signed contributions can cancel.Add the two positive areas.

Area is measured in square units.

Question 4

Find the signed integral and the total shaded area .

The shaded region between the labelled curves and the stated limits.
Check answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -1 to 2, below the axis.Integrate from 2 to 3, above the axis.The signed contributions can cancel.Add the two positive areas.

Area is measured in square units.

Question 5

Find the signed integral and the total shaded area .

The shaded region between the labelled curves and the stated limits.
Check answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -1 to 0, below the axis.Integrate from 0 to 2, above the axis.The signed contributions can cancel.Add the two positive areas.

Area is measured in square units.

← All Calculus cardsReport a mistake