<p>Consider the following region bounded by the curve \(x=g(y)\), the horizontal lines \(y=a\), and \(y = b\), and the \(y\)-axis.</p>
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<p>The area bounded by the curve \(x = \mathrm{g}(y)\), the \(y\)-axis and the lines \(y = a\) and \(y = b\) is given by:</p>
<p style="text-align: center">\[\int_{a}^{b} \mathrm{g}(y) \, \mathrm{d}y \quad \text{or} \quad \int_{a}^{b} x \, \mathrm{d}y\]</p>
<p>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 \(g(y) &lt; 0\) for the interval \([b,a]\).</p>
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<p>Similarly, area \(B_1\) has a positive value while area \(B_2\) has a negative value.</p>
<p>\[\text{Hence, area} = \int_{c}^{b} x \, \mathrm{d}y + \left| \int_{b}^{a} x \, \mathrm{d}y \right|.\]</p>
<p>Similarly, if you are using a GDC, the area above is simply \(\int_{c}^{a} |x| \, \mathrm{d}y\)</p>

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 \(\int_{c}^{a} |x| \, \mathrm{d}y\)

Area of region bounded by curve and y-axis

Area of region bounded by curve and y-axis