Signed Area (VCE SSCE Mathematical Methods): Revision Notes

Signed Area

Understanding signed area

When we work with areas under curves, we need to distinguish between total area and signed area.

Consider the graph of $y = x + 1$ shown below. This line crosses both above and below the $x$-axis, creating two distinct regions.

For this graph:

• Region $A_1$ (above the $x$-axis) has an area of $\frac{1}{2} \times 3 \times 3 = 4\frac{1}{2}$ square units
• Region $A_2$ (below the $x$-axis) has an area of $\frac{1}{2} \times 1 \times 1 = \frac{1}{2}$ square units

The total area is found by adding all regions together:

$$\text{Total area} = A_1 + A_2 = 4\frac{1}{2} + \frac{1}{2} = 5 \text{ square units}$$

The signed area takes into account whether regions are above or below the $x$-axis:

$$\text{Signed area} = A_1 - A_2 = 4\frac{1}{2} - \frac{1}{2} = 4 \text{ square units}$$

Positive and negative signed areas

Here's a more complex example showing multiple regions:

For the shaded regions shown:

• Total area = $A_1 + A_2 + A_3 + A_4$
• Signed area = $A_1 - A_2 + A_3 - A_4$

Notice how we add regions above the $x$-axis and subtract regions below it when calculating signed area.

The definite integral and signed area

This is important to understand: when you evaluate a definite integral, you automatically get the signed area. If part of the graph is below the $x$-axis, that contribution will be negative.

Finding areas in different situations

The method for finding areas depends on whether the function is above or below the $x$-axis in the given interval.

When the function is above the $x$-axis

If f(x) ≥ 0 for all $x \in [a,b]$, then the area of the region between the curve, the $x$-axis, and the vertical lines $x = a$ and $x = b$ is:

$$\text{Area} = \int_a^b f(x)\,dx$$

When the function is below the $x$-axis

If f(x) ≤ 0 for all $x \in [a,b]$, then the area of the region between the curve, the $x$-axis, and the vertical lines $x = a$ and $x = b$ is:

$$\text{Area} = -\int_a^b f(x)\,dx$$

When the function crosses the $x$-axis

If the function crosses the $x$-axis at point $c$ where $a < c < b$, with $f(c) = 0$, $f(x) \geq 0$ for $x \in (c,b]$, and $f(x) \leq 0$ for $x \in [a,c)$, then the total area is:

$$\text{Area} = \int_c^b f(x)\,dx + \left(-\int_a^c f(x)\,dx\right)$$

Worked example: Area below the $x$-axis

Worked example: Areas above and below the $x$-axis

Worked example: Parabola crossing the $x$-axis

Using a CAS calculator

You can verify this result using a TI-Nspire calculator:

Worked example: Cubic function

Properties of the definite integral

These algebraic properties are useful when working with definite integrals:

Quick example

Remember!