Area under curve bounded by x-axis

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).