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Let us first consider in Figure 1 the region bounded by the curve \(y = f(x)\) and the vertical lines \(x=a\) and \(x=b\).

The area bounded by the curve \(y = \mathrm{f}(x)\), the \(x\)-axis and the lines \(x = a\) and \(x = b\) is given by
\[\int_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x \quad \text{or} \quad \int_{a}^{b} y \, \mathrm{d}x \quad \text{where} \, y = \mathrm{f}(x).\]
Let us now consider Figure 2, where the curve \(y=f(x)\) is below the \(x\)-axis.

If the curve is below the \(x\)-axis as in Fig \(2\), then&nbsp;\(\int_{a}^{b} y \, \mathrm{d}x\) is negative.
In this case, the area bounded by the curve \(y=f(x)\) and the vertical lines \(x=a\) and \(x=b\) is given by \[
-\int_a^b \mathrm{f}(x) \, \mathrm{d}x,
\] or equivalently, \[
\left| \int_a^b \mathrm{f}(x) \, \mathrm{d}x \right|.
\]
If the curve \(y = \mathrm{f}(x)\) cuts the \(x\) axis at \(x = b\) as shown in Fig \(3\),
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1772229935168-nfzsla2mngc.png" style="max-width: min(400px, 100%); height: auto; display: block;" alt="FIG 3" class="aligncenter custom-width" /></figure></div>
The area under the curve between \(x = a\) and \(x = c\) is
\[\int_{a}^{b} y \, \mathrm{d}x + \left| \int_{b}^{c} y \, \mathrm{d}x \right|, \]
where&nbsp;\(y = f(x) \leq 0\) on the interval \([b,c]\).
Alternatively, the total area between \(x = a\) and \(x = c\) is \(\int_{a}^{c} |\mathrm{f}(x)| \, \mathrm{d}x\).
This is especially useful if you are working out the area using a Graphic Display Calculator (GDC) in attempting questions in Paper 2, where only numerical value of definite integrals are required (and not the exact values).

Let us first consider in Figure 1 the region bounded by the curve \(y = f(x)\) and the vertical lines \(x=a\) and \(x=b\).

The area bounded by the curve \(y = \mathrm{f}(x)\), the \(x\)-axis and the lines \(x = a\) and \(x = b\) is given by

\[\int_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x \quad \text{or} \quad \int_{a}^{b} y \, \mathrm{d}x \quad \text{where} \, y = \mathrm{f}(x).\]

Let us now consider Figure 2, where the curve \(y=f(x)\) is below the \(x\)-axis.

If the curve is below the \(x\)-axis as in Fig \(2\), then \(\int_{a}^{b} y \, \mathrm{d}x\) is negative.

In this case, the area bounded by the curve \(y=f(x)\) and the vertical lines \(x=a\) and \(x=b\) is given by
\[
-\int_a^b \mathrm{f}(x) \, \mathrm{d}x,
\]
or equivalently,
\[
\left| \int_a^b \mathrm{f}(x) \, \mathrm{d}x \right|.
\]

If the curve \(y = \mathrm{f}(x)\) cuts the \(x\) axis at \(x = b\) as shown in **Fig \(3\)**,

The area under the curve between \(x = a\) and \(x = c\) is

\[\int_{a}^{b} y \, \mathrm{d}x + \left| \int_{b}^{c} y \, \mathrm{d}x \right|, \]

where \(y = f(x) \leq 0\) on the interval \([b,c]\).

Alternatively, the total area between \(x = a\) and \(x = c\) is \(\int_{a}^{c} |\mathrm{f}(x)| \, \mathrm{d}x\).

This is especially useful if you are working out the area using a Graphic Display Calculator (GDC) in attempting questions in Paper 2, where only numerical value of definite integrals are required (and not the exact values).

Area under curve bounded by x-axis

Area under curve bounded by x-axis