QuestionThe diagram show part of the curve $y=3-\frac{18}{2 x+3}$. The normal to the curve at $x=3$ intersects the $y$-axis at $A$. Find the exact area of the shaded region.

Studdy Solution
Find the exact area of the shaded region by subtracting the area under the normal line from the area under the curve.
$$\text{Area}_{\text{shaded}} = \left( 9 - 9 \ln(3) \right) - \left( -\frac{57}{8} \right)$$ $$= 9 - 9 \ln(3) + \frac{57}{8}$$ $$= \frac{72}{8} - 9 \ln(3) + \frac{57}{8}$$ $$= \frac{129}{8} - 9 \ln(3)$$
Final Answer: The exact area of the shaded region is:
$$\boxed{\frac{129}{8} - 9 \ln(3)}$$