1. The problem is to find the area under the curve of the function $$y = -0.015x^2 + 60$$ over a certain interval. 2. To find the area under a curve, we use the definite integral of the function over the interval of interest. The formula is: $$\text{Area} = \int_a^b y \, dx = \int_a^b (-0.015x^2 + 60) \, dx$$ 3. Since you want to model a pencil skirt, typically the domain for $x$ could represent the length of the skirt. Let's assume the skirt length ranges from $x=0$ to $x=40$ (units could be cm or inches depending on your model). 4. Compute the definite integral: $$\int_0^{40} (-0.015x^2 + 60) \, dx = \left[-0.015 \frac{x^3}{3} + 60x \right]_0^{40}$$ 5. Simplify the integral expression: $$= \left[-0.005x^3 + 60x \right]_0^{40}$$ 6. Evaluate at the bounds: $$= \left(-0.005 \times 40^3 + 60 \times 40\right) - \left(-0.005 \times 0 + 60 \times 0\right)$$ $$= (-0.005 \times 64000) + 2400 - 0$$ $$= -320 + 2400 = 2080$$ 7. So, the area under the curve from $x=0$ to $x=40$ is $2080$ square units. 8. Regarding modeling skirts, the function $y = -0.015x^2 + 60$ is a downward-opening parabola with a maximum at $x=0$ and decreasing as $x$ increases, which could represent the narrowing shape of a pencil skirt. This model could be a good mathematical representation for the skirt's silhouette in your math IA, especially if you interpret $x$ as the vertical length and $y$ as the width at that height.