Question

Find the area of the region bounded by the curve xy − 3x − 2y − 10 = 0, x-axis and the lines x = 3, x = 4.

Answer 1

We have,
$$\begin{gathered} xy-3x-2y-10=0 \\ \Rightarrow xy-2y=3x+10 \\ \Rightarrow y\left(x-2\right)=3x+10 \\ \Rightarrow y=\frac{3x+10}{x-2} \end{gathered}$$
Let A represent the required area:
$$\begin{gathered} \Rightarrow A={\int }_3^4\left|y\right|\mathit{d}x \\ ={\int }_3^4\frac{3x+10}{x-2}\mathit{d}x \\ ={\int }_3^4\frac{3x-6+16}{x-2}\mathit{d}x \\ ={\int }_3^4\left(3+\frac{16}{x-2}\right)\mathit{d}x \\ ={\left[3x+16\log \left|x-2\right|\right]}_3^4 \\ =\left[12+16\log \left|2\right|-9-16\log \left|1\right|\right] \\ =3+16\log 2\mathrm{sq}.\mathrm{units} \end{gathered}$$