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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.
Solution
We have,
\[xy - 3x - 2y - 10 = 0\]
\[ \Rightarrow xy - 2y = 3x + 10\]
\[ \Rightarrow y\left( x - 2 \right) = 3x + 10\]
\[ \Rightarrow y = \frac{3x + 10}{x - 2}\]
Let A represent the required area:
\[\Rightarrow A = \int_3^4 \left| y \right| d x\]
\[ = \int_3^4 \frac{3x + 10}{x - 2} d x\]
\[ = \int_3^4 \frac{3x - 6 + 16}{x - 2} d x\]
\[ = \int_3^4 \left( 3 + \frac{16}{x - 2} \right) d x\]
\[ = \left[ 3x + 16 \log \left| x - 2 \right| \right]_3^4 \]
\[ = \left[ 12 + 16 \log \left| 2 \right| - 9 - 16 \log \left| 1 \right| \right]\]
\[ = 3 + 16 \log 2\text{ sq . units }\]
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