Area of region bounded by curve and y-axis

Consider the following region bounded by the curve \(x=g(y)\), the horizontal lines \(y=a\), and \(y = b\), and the \(y\)-axis.

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

\[\int_{a}^{b} \mathrm{g}(y) \, \mathrm{d}y \quad \text{or} \quad \int_{a}^{b} x \, \mathrm{d}y\]

Now, let us consider another region bounded by the curve \(x=g(y)\), the horizontal lines \(y=a\) and \(y=b\), and the \(y\)-axis, but this time \(g(y) < 0\) for the interval \([b,a]\).

Similarly, area \(B_1\) has a positive value while area \(B_2\) has a negative value.

\[\text{Hence, area} = \int_{c}^{b} x \, \mathrm{d}y + \left| \int_{b}^{a} x \, \mathrm{d}y \right|.\]

Similarly, if you are using a GDC, the area above is simply \(\int_{c}^{a} |x| \, \mathrm{d}y\)