The graph of $y = -x^2 - 9$ is shown below - AQA - A-Level Maths Pure - Question 1 - 2019 - Paper 3

To find the area of the shaded region under the curve of the function, we need to first determine the points of intersection between the graph and the x-axis.

1. Determine the points of intersection: The equation of the graph is given by:

   $y = -x^2 - 9$ Setting this to 0 to find the x-intercepts:

   $0 = -x^2 - 9$

   This simplifies to:

   $x^2 = -9$

   Since there are no real solutions, the parabola does not intersect the x-axis. Thus, the area of the shaded region would be calculated over the range where the graph is above the x-axis from the vertex.

2. Calculate the area under the curve: The area can be determined using the definite integral from the vertex x-coordinate (which is zero) to the extent of the graph. The vertex occurs at:

   $x = 0$

   This indicates the calculations should represent the enclosed area below the vertex.

3. Setting up the integral: The area can be represented as:

   A = -igg[rac{(x^3)}{3} + 9xigg] \text{ from } 0 \text{ to } 3

   Evaluating this integral will yield the required area, which can be computed to be:

   $A + 9*3 = 27$ (Triangle area under the shape from vertex to closures at 3)

4. Final Result: Thus, the final area of the shaded region is 18 square units. Remember to circle this answer as per the question's instructions.